FunctorsFrom Wikipedia, the free encyclopediaContents1 Adjoint functors 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Spelling (or morphology) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Solutions to optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Symmetry of optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Conventions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Universal morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 Counit-unit adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 Hom-set adjunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Adjunctions in full . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4.1 Universal morphisms induce hom-set adjunction . . . . . . . . . . . . . . . . . . . . . . . 61.4.2 Counit-unit adjunction induces hom-set adjunction. . . . . . . . . . . . . . . . . . . . . . 61.4.3 Hom-set adjunction induces all of the above . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.1 Ubiquity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.2 Problems formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5.3 Posets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6.1 Free groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6.2 Free constructions and forgetful functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6.3 Diagonal functors and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6.4 Colimits and diagonal functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.6.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.7 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7.3 Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7.4 Limit preservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.7.5 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8 Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15iii CONTENTS1.8.1 Universal constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8.2 Equivalences of categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.8.3 Monads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 Amnestic functor 172.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Polynomial functor 183.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 Polynomial functor 204.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.3 Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.6 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Derived functor 225.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.2 Construction and rst properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225.3 Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.5 Naturality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.6 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 Diagonal functor 267 Diagram (category theory) 277.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277.3 Cones and limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.4 Commutative diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29CONTENTS iii8 Dinatural transformation 308.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308.2 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Dominant functor 329.1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210Eaceable functor 3310.1Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3310.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3311End (category theory) 3411.1Coend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3511.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3612Essentially surjective functor 3713Exact functor 3813.1Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.3Properties and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.4Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4013.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014Final functor 4114.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4114.2See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4115Forgetful functor 4215.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.2Left Adjoint: Free. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4316Free functor 4416.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4416.2.1 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4516.3Free universal algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.4Free functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.4.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.4.2 General case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4616.5List of free objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46iv CONTENTS16.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4716.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4717Full and faithful functors 4817.1Formal denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4817.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4917.6References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4918Functor 5018.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.1.1 Covariance and contravariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5018.1.2 Opposite functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.1.3 Bifunctors and multifunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5118.3Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5218.4Relation to other categorical concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.5Computer implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5318.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5418.9External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5419Functor category 5519.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5519.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5519.3Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5619.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5719.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5720GabrielPopescu theorem 5820.1Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5820.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5820.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5921Hom functor 6021.1Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6021.2Yonedas lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6121.3Internal Hom functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6121.4Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6221.5Other properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62CONTENTS v21.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6221.7Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6221.8References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6322Inverse system 6422.1The category of inverse systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6422.2Direct systems/Ind-objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6422.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6422.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6522.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6523Logical functor 6623.1Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6623.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6624Monoidal category action 6724.1References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6725Monoidal functor 6825.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6825.1.1 Variants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6925.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6925.3Monoidal functors and adjunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7025.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7025.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7026Natural transformation 7126.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7126.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7226.2.1 Opposite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7226.2.2 Double dual of a vector space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7226.2.3 Tensor-hom adjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7226.3Unnatural isomorphism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7326.3.1 Example: fundamental group of torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7326.3.2 Example: dual of a nite-dimensional vector space . . . . . . . . . . . . . . . . . . . . . . 7426.4Operations with natural transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7426.5Functor categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7526.6Yoneda lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7526.7Historical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7526.8See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7626.9Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7626.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7626.11External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76vi CONTENTS27Polynomial functor 7727.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7727.2Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7727.3Branches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7827.4History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7827.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7827.6External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7828Presheaf (category theory) 7928.1Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7928.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7928.3See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7928.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8029Profunctor 8129.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8129.1.1 Composition of profunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8129.1.2 The bicategory of profunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8229.2Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8229.2.1 Lifting functors to profunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8229.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8229.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8230Pseudofunctor 8330.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8330.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8330.3External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8331Representable functor 8431.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8431.2Universal elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8431.3Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8531.4Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8531.4.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8531.4.2 Preservation of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8631.4.3 Left adjoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8631.5Relation to universal morphisms and adjoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8631.6See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8631.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8632Simplicial presheaf 8732.1Homotopy sheaves of a simplicial presheaf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8732.2Model structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87CONTENTS vii32.3Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8832.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8832.5Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8832.6Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8832.7References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8832.8External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8833Simplicial set 8933.1Motivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8933.2Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8933.3Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9033.4Face and degeneracy maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9033.5Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9033.6The standard n-simplex and the category of simplices . . . . . . . . . . . . . . . . . . . . . . . . . 9133.7Geometric realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9133.8Singular set for a space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.9Homotopy theory of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.10Simplicial objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9233.11History and uses of simplicial sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.12See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.13Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9333.14References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9334Smooth functor 9534.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9534.2Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9534.3References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9635Span (category theory) 9735.1Formal denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9735.2Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9735.3Cospans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9735.4See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9835.5References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9836Stone functor 9936.1See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9936.2References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9937Subfunctor 10037.1Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10037.2Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10037.3Open subfunctors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100viii CONTENTS38Zuckerman functor 10238.1Notation and terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10238.2Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10238.3Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10238.4References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10238.5Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 10438.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10438.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10638.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Chapter 1Adjoint functorsFor the construction in eld theory, see Adjunction (eld theory). For the construction in topology, see Adjunctionspace.In mathematics, specically category theory, adjunction is a possible relationship between two functors.Adjunction is ubiquitous in mathematics, as it species intuitive notions of optimization and eciency.In the most concise symmetric denition, an adjunction between categories C and D is a pair of functors,F: D C and G : C Dand a family of bijectionshomC(FY, X) = homD(Y, GX)which is natural in the variables X and Y. The functor F is called a left adjoint functor, while G is called a rightadjoint functor. The relationship F is left adjoint to G (or equivalently, G is right adjoint to F) is sometimeswrittenF G.This denition and others are made precise below.1.1 IntroductionThe slogan is Adjoint functors arise everywhere. (Saunders Mac Lane, Categories for the working mathematician)The long list of examples in this article is only a partial indication of how often an interesting mathematical construc-tion is an adjoint functor. As a result, general theorems about left/right adjoint functors, such as the equivalence oftheir various denitions or the fact that they respectively preserve colimits/limits (which are also found in every areaof mathematics), can encode the details of many useful and otherwise non-trivial results.1.1.1 Spelling (or morphology)One can observe (e.g. in this article), two dierent roots are used: adjunct and adjoint. From Oxford shorterEnglish dictionary, adjunct is from Latin, adjoint is from French.In Mac Lane, Categories for the working mathematician, chap. 4, Adjoints, one can verify the following usage. : homC(FY, X) = homD(Y, GX)12 CHAPTER 1. ADJOINT FUNCTORSThe hom-set bijection is an adjunction.If f an arrow in homC(FY, X) , f is the right adjunct of f (p. 81).The functor F is left adjoint for G .1.2 Motivation1.2.1 Solutions to optimization problemsIt can be said that an adjoint functor is a way of giving the most ecient solution to some problem via a method whichis formulaic. For example, an elementary problem in ring theory is how to turn a rng (which is like a ring that mightnot have a multiplicative identity) into a ring. The most ecient way is to adjoin an element '1' to the rng, adjoinall (and only) the elements which are necessary for satisfying the ring axioms (e.g. r+1 for each r in the ring), andimpose no relations in the newly formed ring that are not forced by axioms. Moreover, this construction is formulaicin the sense that it works in essentially the same way for any rng.This is rather vague, though suggestive, and can be made precise in the language of category theory: a constructionis most ecient if it satises a universal property, and is formulaic if it denes a functor. Universal properties comein two types: initial properties and terminal properties. Since these are dual (opposite) notions, it is only necessaryto discuss one of them.The idea of using an initial property is to set up the problem in terms of some auxiliary category E, and then identifythat what we want is to nd an initial object of E. This has an advantage that the optimization the sense that weare nding the most ecient solution means something rigorous and is recognisable, rather like the attainment ofa supremum. The category E is also formulaic in this construction, since it is always the category of elements of thefunctor to which one is constructing an adjoint. In fact, this latter category is precisely the comma category over thefunctor in question.As an example, take the given rng R, and make a category E whose objects are rng homomorphisms R S, with S aring having a multiplicative identity. The morphisms in E between R S1 and R S2 are commutative triangles ofthe form (R S1,R S2, S1 S2) where S1 S2 is a ring map (which preserves the identity). Note that this isprecisely the denition of the comma category of R over the inclusion of unitary rings into rng. The existence of amorphism between R S1 and R S2 implies that S1 is at least as ecient a solution as S2 to our problem: S2 canhave more adjoined elements and/or more relations not imposed by axioms than S1. Therefore, the assertion that anobject R R* is initial in E, that is, that there is a morphism from it to any other element of E, means that the ringR* is a most ecient solution to our problem.The two facts that this method of turning rngs into rings is most ecient and formulaic can be expressed simultaneouslyby saying that it denes an adjoint functor.1.2.2 Symmetry of optimization problemsContinuing this discussion, suppose we started with the functor F, and posed the following (vague) question: is therea problem to which F is the most ecient solution?The notion that F is the most ecient solution to the problem posed by G is, in a certain rigorous sense, equivalent tothe notion that G poses the most dicult problem that F solves.This has the intuitive meaning that adjoint functors should occur in pairs, and in fact they do, but this is not trivial fromthe universal morphism denitions. The equivalent symmetric denitions involving adjunctions and the symmetriclanguage of adjoint functors (we can say either F is left adjoint to G or G is right adjoint to F) have the advantage ofmaking this fact explicit.1.3 Formal denitionsThere are various denitions for adjoint functors. Their equivalence is elementary but not at all trivial and in facthighly useful. This article provides several such denitions:1.3. FORMAL DEFINITIONS 3The denitions via universal morphisms are easy to state, and require minimal verications when constructingan adjoint functor or proving two functors are adjoint. They are also the most analogous to our intuitioninvolving optimizations.The denition via counit-unit adjunction is convenient for proofs about functors which are known to be adjoint,because they provide formulas that can be directly manipulated.The denition via hom-sets makes symmetry the most apparent, and is the reason for using the word adjoint.Adjoint functors arise everywhere, in all areas of mathematics. Their full usefulness lies in that the structure in anyof these denitions gives rise to the structures in the others via a long but trivial series of deductions. Thus, switchingbetween themmakes implicit use of a great deal of tedious details that would otherwise have to be repeated separatelyin every subject area. For example, naturality and terminality of the counit can be used to prove that any right adjointfunctor preserves limits.1.3.1 ConventionsThe theory of adjoints has the terms left and right at its foundation, and there are many components which live in oneof two categories C and D which are under consideration. It can therefore be extremely helpful to choose letters inalphabetical order according to whether they live in the lefthand category C or the righthand category D, and alsoto write them down in this order whenever possible.In this article for example, the letters X, F, f, will consistently denote things which live in the category C, the lettersY, G, g, will consistently denote things which live in the category D, and whenever possible such things will bereferred to in order from left to right (a functor F:CD can be thought of as living where its outputs are, in C).1.3.2 Universal morphismsA functor F : C D is a left adjoint functor if for each object X in C, there exists a terminal morphism from F toX. If, for each object X in C, we choose an object G0X of D for which there is a terminal morphism X : F(G0X) X from F to X, then there is a unique functor G : C D such that GX = G0X and X FG(f) = f X for f : X X a morphism in C; F is then called a left adjoint toG.A functor G : C D is a right adjoint functor if for each object Y in D, there exists an initial morphism from Y toG. If, for each object Y in D, we choose an object F0Y of C and an initial morphism Y : Y G(F0Y) from Y to G,then there is a unique functor F : C D such that FY = F0Y and GF(g) Y = Y g for g : Y Y a morphismin D; G is then called a right adjoint toF.Remarks:It is true, as the terminology implies, that F is left adjoint to G if and only if G is right adjoint to F. This is apparent fromthe symmetric denitions given below. The denitions via universal morphisms are often useful for establishing thata given functor is left or right adjoint, because they are minimalistic in their requirements. They are also intuitivelymeaningful in that nding a universal morphism is like solving an optimization problem.1.3.3 Counit-unit adjunctionA counit-unit adjunction between two categories C and D consists of two functors F : C D and G : C D andtwo natural transformations : FG 1C: 1D GFrespectively called thecounit and theunit of the adjunction (terminology from universal algebra), such that thecompositionsFFFGFFF4 CHAPTER 1. ADJOINT FUNCTORSGGGFGGGare the identity transformations 1F and 1G on F and G respectively.In this situation we say that F is left adjoint to GandG is right adjoint to F , and may indicate this relationshipby writing (, ) : F G , or simply F G .In equation form, the above conditions on (,) are the counit-unit equations1F= F F1G= G Gwhich mean that for each X in C and each Y in D,1FY= FY F(Y )1GX= G(X) GXNote that here 1 denotes identity functors, while above the same symbol was used for identity natural transformations.These equations are useful in reducing proofs about adjoint functors to algebraic manipulations. They are sometimescalled the zig-zag equations because of the appearance of the corresponding string diagrams. A way to rememberthem is to rst write down the nonsensical equation 1 = and then ll in either F or G in one of the two simpleways which make the compositions dened.Note: The use of the prex co in counit here is not consistent with the terminology of limits and colimits, becausea colimit satises an initial property whereas the counit morphisms will satisfy terminal properties, and dually. Theterm unit here is borrowed from the theory of monads where it looks like the insertion of the identity 1 into a monoid.1.3.4 Hom-set adjunctionA hom-set adjunction between two categories C and D consists of two functors F : C D and G : C D and anatural isomorphism : homC(F, ) homD(, G)This species a family of bijectionsY,X: homC(FY, X) homD(Y, GX)for all objects X in C and Y in D.In this situation we say that F is left adjoint to GandG is right adjoint to F , and may indicate this relationshipby writing : F G , or simply F G .This denition is a logical compromise in that it is somewhat more dicult to satisfy than the universal morphismdenitions, and has fewer immediate implications than the counit-unit denition. It is useful because of its obvioussymmetry, and as a stepping-stone between the other denitions.In order to interpret as a natural isomorphism, one must recognize homC(F, ) and homD(, G) as functors. Infact, they are both bifunctors from Dop C to Set (the category of sets). For details, see the article on hom functors.Explicitly, the naturality of means that for all morphisms f : X X in C and all morphisms g : Y Y in D thefollowing diagram commutes:The vertical arrows in this diagramare those induced by composition with f and g. Formally, Hom(Fg, f) : HomC(FY,X) HomC(FY, X ) is given by h f o h o Fg for each h in HomC(FY, X). Hom(g, Gf) is similar.1.4 Adjunctions in fullThere are hence numerous functors and natural transformations associated with every adjunction, and only a smallportion is sucient to determine the rest.1.4. ADJUNCTIONS IN FULL 5Naturality of An adjunction between categories C and D consists ofA functor F : C D called the left adjointA functor G : C D called the right adjointA natural isomorphism : homC(F,) homD(,G)A natural transformation : FG 1C called the counitA natural transformation : 1D GF called the unitAn equivalent formulation, where X denotes any object of C and Y denotes any object of D:For every C-morphism f : FY X, there is a unique D-morphism Y, X(f) = g : Y GX such that the diagramsbelow commute, and for every D-morphism g : Y GX, there is a unique C-morphism 1Y, X(g) = f : FY X inC such that the diagrams below commute:From this assertion, one can recover that:6 CHAPTER 1. ADJOINT FUNCTORSThe transformations , , and are related by the equationsf= 1Y,X(g) = X F(g) homC(F(Y ), X)g= Y,X(f) = G(f) Y homD(Y, G(X))1GX,X(1GX) = X homC(FG(X), X)Y,FY (1FY ) = Y homD(Y, GF(Y ))The transformations , satisfy the counit-unit equations1FY= FY F(Y )1GX= G(X) GXEach pair (GX, X) is a terminal morphism from F to X in CEach pair (FY, Y) is an initial morphism from Y to G in DIn particular, the equations above allow one to dene , , and in terms of any one of the three. However, theadjoint functors F and G alone are in general not sucient to determine the adjunction. We will demonstrate theequivalence of these situations below.1.4.1 Universal morphisms induce hom-set adjunctionGiven a right adjoint functor G : C D; in the sense of initial morphisms, one may construct the induced hom-setadjunction by doing the following steps.Construct a functor F : C D and a natural transformation .For each object Yin D, choose an initial morphism (F(Y), Y) from Yto G, so we have Y: YG(F(Y)). We have the map of F on objects and the family of morphisms .For each f : Y0 Y1, as (F(Y0), Y0) is an initial morphism, then factorize Y1 o f with Y0 and getF(f) : F(Y0) F(Y1). This is the map of F on morphisms.The commuting diagram of that factorization implies the commuting diagram of natural transformations,so : 1D G o F is a natural transformation.Uniqueness of that factorization and that G is a functor implies that the map of F on morphisms preservescompositions and identities.Construct a natural isomorphism : homC(F-,-) homD(-,G-).For each object X in C, each object Y in D, as (F(Y), Y) is an initial morphism, then Y, X is a bijection,where Y, X(f : F(Y) X) = G(f) o Y. is a natural transformation, G is a functor, then for any objects X0, X1 in C, any objects Y0, Y1 in D,any x : X0 X1, any y : Y1 Y0, we have Y1, X1(x o f o F(y)) = G(x) o G(f) o G(F(y)) o Y1 =G(x) o G(f) o Y0 o y = G(x) o Y0, X0(f) o y, and then is natural in both arguments.Asimilar argument allows one to construct a hom-set adjunction fromthe terminal morphisms to a left adjoint functor.(The construction that starts with a right adjoint is slightly more common, since the right adjoint in many adjoint pairsis a trivially dened inclusion or forgetful functor.)1.4.2 Counit-unit adjunction induces hom-set adjunctionGiven functors F : C D, G : C D, and a counit-unit adjunction (, ) : F G, we can construct a hom-setadjunction by nding the natural transformation : homC(F-,-) homD(-,G-) in the following steps:For each f : FY X and each g : Y GX, dene1.4. ADJUNCTIONS IN FULL 7Y,X(f) = G(f) YY,X(g) = X F(g)The transformations and are natural because and are natural.Using, in order, that F is a functor, that is natural, and the counit-unit equation 1FY = FY o F(Y), we obtainf= X FG(f) F(Y )= f FY F(Y )= f 1FY= fhence is the identity transformation.Dually, using that G is a functor, that is natural, and the counit-unit equation 1GX = G(X) o GX, we obtaing= G(X) GF(g) Y= G(X) GX g= 1GX g= ghence is the identity transformation. Thus is a natural isomorphism with inverse 1 = .1.4.3 Hom-set adjunction induces all of the aboveGiven functors F : C D, G : C D, and a hom-set adjunction : homC(F-,-) homD(-,G-), we can construct acounit-unit adjunction(, ) : F G ,which denes families of initial and terminal morphisms, in the following steps:Let X= 1GX,X(1GX) homC(FGX, X) for each X in C, where 1GX homD(GX, GX) is the identitymorphism.Let Y=Y,FY (1FY ) homD(Y, GFY ) for each Y in D, where 1FY homC(FY, FY ) is the identitymorphism.The bijectivity and naturality of imply that each (GX, X) is a terminal morphism from F to X in C, andeach (FY, Y) is an initial morphism from Y to G in D.The naturality of implies the naturality of and , and the two formulasY,X(f) = G(f) Y1Y,X(g) = X F(g)for each f: FY X and g: Y GX (which completely determine ).Substituting FY for X and Y = Y, FY(1FY) for g in the second formula gives the rst counit-unit equation1FY= FY F(Y ) ,and substituting GX for Y and X= 1GX, X(1GX) for f in the rst formula gives the second counit-unitequation1GX= G(X) GX .8 CHAPTER 1. ADJOINT FUNCTORS1.5 History1.5.1 UbiquityThe idea of an adjoint functor was formulated by Daniel Kan in 1958. Like many of the concepts in category theory,it was suggested by the needs of homological algebra, which was at the time devoted to computations. Those facedwith giving tidy, systematic presentations of the subject would have noticed relations such ashom(F(X), Y) = hom(X, G(Y))in the category of abelian groups, where F was the functor A (i.e. take the tensor product with A), and G wasthe functor hom(A,). The use of the equals sign is an abuse of notation; those two groups are not really identicalbut there is a way of identifying them that is natural. It can be seen to be natural on the basis, rstly, that these aretwo alternative descriptions of the bilinear mappings from X A to Y. That is, however, something particular to thecase of tensor product. In category theory the 'naturality' of the bijection is subsumed in the concept of a naturalisomorphism.The terminology comes from the Hilbert space idea of adjoint operators T, U with Tx, y= x, Uy , which isformally similar to the above relation between hom-sets. We say that F is left adjoint to G, and G is right adjoint toF. Note that G may have itself a right adjoint that is quite dierent from F (see below for an example). The analogyto adjoint maps of Hilbert spaces can be made precise in certain contexts.[1]If one starts looking for these adjoint pairs of functors, they turn out to be very common in abstract algebra, andelsewhere as well. The example section below provides evidence of this; furthermore, universal constructions, whichmay be more familiar to some, give rise to numerous adjoint pairs of functors.In accordance with the thinking of Saunders Mac Lane, any idea, such as adjoint functors, that occurs widely enoughin mathematics should be studied for its own sake.1.5.2 Problems formulationsMathematicians do not generally need the full adjoint functor concept. Concepts can be judged according to theiruse in solving problems, as well as for their use in building theories. The tension between these two motivations wasespecially great during the 1950s when category theory was initially developed. Enter Alexander Grothendieck, whoused category theory to take compass bearings in other work in functional analysis, homological algebra and nallyalgebraic geometry.It is probably wrong to say that he promoted the adjoint functor concept in isolation: but recognition of the role ofadjunction was inherent in Grothendiecks approach. For example, one of his major achievements was the formulationof Serre duality in relative form loosely, in a continuous family of algebraic varieties. The entire proof turned onthe existence of a right adjoint to a certain functor. This is something undeniably abstract, and non-constructive, butalso powerful in its own way.1.5.3 PosetsEvery partially ordered set can be viewed as a category (with a single morphism between x and y if and only if x y).A pair of adjoint functors between two partially ordered sets is called a Galois connection (or, if it is contravariant,an antitone Galois connection). See that article for a number of examples: the case of Galois theory of course is aleading one. Any Galois connection gives rise to closure operators and to inverse order-preserving bijections betweenthe corresponding closed elements.As is the case for Galois groups, the real interest lies often in rening a correspondence to a duality (i.e. antitoneorder isomorphism). A treatment of Galois theory along these lines by Kaplansky was inuential in the recognitionof the general structure here.The partial order case collapses the adjunction denitions quite noticeably, but can provide several themes:adjunctions may not be dualities or isomorphisms, but are candidates for upgrading to that status1.6. EXAMPLES 9closure operators may indicate the presence of adjunctions, as corresponding monads (cf. the Kuratowskiclosure axioms)a very general comment of William Lawvere[2] is that syntax and semantics are adjoint: take C to be the setof all logical theories (axiomatizations), and D the power set of the set of all mathematical structures. For atheory T in C, let F(T) be the set of all structures that satisfy the axioms T; for a set of mathematical structuresS, let G(S) be the minimal axiomatization of S. We can then say that F(T) is a subset of S if and only if Tlogically implies G(S): the semantics functor F is left adjoint to the syntax functor G.division is (in general) the attempt to invert multiplication, but many examples, such as the introduction ofimplication in propositional logic, or the ideal quotient for division by ring ideals, can be recognised as theattempt to provide an adjoint.Together these observations provide explanatory value all over mathematics.1.6 Examples1.6.1 Free groupsThe construction of free groups is a common and illuminating example.Suppose that F : Grp Set is the functor assigning to each set Y the free group generated by the elements of Y, andthat G : Grp Set is the forgetful functor, which assigns to each group X its underlying set. Then F is left adjointto G:Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements ofX. Let X: FGX X be the group homomorphism which sends the generators of FGX to the elements of X theycorrespond to, which exists by the universal property of free groups. Then each (GX, X) is a terminal morphismfrom F to X, because any group homomorphism from a free group FZ to X will factor through X: FGX X viaa unique set map from Z to GX. This means that (F,G) is an adjoint pair.Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. LetY: Y GFYbe the set map given by inclusion of generators. Then each (FY, Y ) is an initial morphism fromY to G, because any set map from Y to the underlying set GW of a group will factor through Y: Y GFYvia aunique group homomorphism from FY to W. This also means that (F,G) is an adjoint pair.Hom-set adjunction. Maps from the free group FY to a group X correspond precisely to maps from the set Y to theset GX: each homomorphism from FY to X is fully determined by its action on generators. One can verify directlythat this correspondence is a natural transformation, which means it is a hom-set adjunction for the pair (F,G).Counit-unit adjunction. One can also verify directly that and are natural. Then, a direct verication that theyform a counit-unit adjunction (, ) : F G is as follows:The rst counit-unit equation 1F= F F says that for each set Y the compositionFYF(Y)FGFYFYFYshould be the identity. The intermediate group FGFYis the free group generated freely by the words of the freegroup FY. (Think of these words as placed in parentheses to indicate that they are independent generators.) Thearrow F(Y ) is the group homomorphism from FY into FGFY sending each generator y of FY to the correspondingword of length one (y) as a generator of FGFY. The arrowFYis the group homomorphismfromFGFY to FY sendingeach generator to the word of FY it corresponds to (so this map is dropping parentheses). The composition of thesemaps is indeed the identity on FY.The second counit-unit equation 1G= G G says that for each group X the compositionGXGXGFGXG(X)GXshould be the identity. The intermediate set GFGX is just the underlying set of FGX. The arrow GX is the inclusionof generators set map from the set GX to the set GFGX. The arrow G(X) is the set map from GFGX to GX which10 CHAPTER 1. ADJOINT FUNCTORSunderlies the group homomorphism sending each generator of FGX to the element of X it corresponds to (droppingparentheses). The composition of these maps is indeed the identity on GX.1.6.2 Free constructions and forgetful functorsFree objects are all examples of a left adjoint to a forgetful functor which assigns to an algebraic object its underlyingset. These algebraic free functors have generally the same description as in the detailed description of the free groupsituation above.1.6.3 Diagonal functors and limitsProducts, bred products, equalizers, and kernels are all examples of the categorical notion of a limit. Any limitfunctor is right adjoint to a corresponding diagonal functor (provided the category has the type of limits in question),and the counit of the adjunction provides the dening maps from the limit object (i.e. from the diagonal functor onthe limit, in the functor category). Below are some specic examples.Products Let : Grp2 Grp the functor which assigns to each pair (X1, X2) the product group X1X2,and let : Grp2 Grp be the diagonal functor which assigns to every group X the pair (X, X) in the productcategory Grp2. The universal property of the product group shows that is right-adjoint to . The counit ofthis adjunction is the dening pair of projection maps from X1X2 to X1 and X2 which dene the limit, andthe unit is the diagonal inclusion of a group X into X1X2 (mapping x to (x,x)).The cartesian product of sets, the product of rings, the product of topological spaces etc. follow thesame pattern; it can also be extended in a straightforward manner to more than just two factors. Moregenerally, any type of limit is right adjoint to a diagonal functor.Kernels. Consider the category D of homomorphisms of abelian groups. If f1 : A1 B1 and f2 : A2 B2are two objects of D, then a morphism from f1 to f2 is a pair (gA, gB) of morphisms such that gBf1 = f2gA.Let G : D Ab be the functor which assigns to each homomorphism its kernel and let F : D Ab be thefunctor which maps the group Ato the homomorphismA0. Then G is right adjoint to F, which expresses theuniversal property of kernels. The counit of this adjunction is the dening embedding of a homomorphismskernel into the homomorphisms domain, and the unit is the morphism identifying a group A with the kernelof the homomorphism A 0.A suitable variation of this example also shows that the kernel functors for vector spaces and for modulesare right adjoints. Analogously, one can showthat the cokernel functors for abelian groups, vector spacesand modules are left adjoints.1.6.4 Colimits and diagonal functorsCoproducts, bred coproducts, coequalizers, and cokernels are all examples of the categorical notion of a colimit.Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimitsin question), and the unit of the adjunction provides the dening maps into the colimit object. Beloware some specicexamples.Coproducts. If F : Ab Ab2assigns to every pair (X1, X2) of abelian groups their direct sum, and if G :Ab Ab2is the functor which assigns to every abelian group Y the pair (Y, Y), then F is left adjoint to G,again a consequence of the universal property of direct sums. The unit of this adjoint pair is the dening pairof inclusion maps from X1 and X2 into the direct sum, and the counit is the additive map from the direct sumof (X,X) to back to X (sending an element (a,b) of the direct sum to the element a+b of X).Analogous examples are given by the direct sum of vector spaces and modules, by the free product ofgroups and by the disjoint union of sets.1.6. EXAMPLES 111.6.5 Further examplesAlgebraAdjoining an identity to a rng. This example was discussed in the motivation section above. Given a rng R,a multiplicative identity element can be added by taking RxZ and dening a Z-bilinear product with (r,0)(0,1)= (0,1)(r,0) = (r,0), (r,0)(s,0) = (rs,0), (0,1)(0,1) = (0,1). This constructs a left adjoint to the functor taking aring to the underlying rng.Ring extensions. Suppose R and S are rings, and : R S is a ring homomorphism. Then S can be seen asa (left) R-module, and the tensor product with S yields a functor F : R-Mod S-Mod. Then F is left adjointto the forgetful functor G : S-Mod R-Mod.Tensor products. If R is a ring and M is a right R module, then the tensor product with M yields a functor F: R-Mod Ab. The functor G : Ab R-Mod, dened by G(A) = homZ(M,A) for every abelian group A, isa right adjoint to F.From monoids and groups to rings The integral monoid ring construction gives a functor from monoidsto rings. This functor is left adjoint to the functor that associates to a given ring its underlying multiplicativemonoid. Similarly, the integral group ring construction yields a functor from groups to rings, left adjoint to thefunctor that assigns to a given ring its group of units. One can also start with a eld K and consider the categoryof K-algebras instead of the category of rings, to get the monoid and group rings over K.Field of fractions. Consider the category Dom of integral domains with injective morphisms. The forgetfulfunctor Field Dom from elds has a left adjoint - it assigns to every integral domain its eld of fractions.Polynomial rings. Let Ring* be the category of pointed commutative rings with unity (pairs (A,a) where Ais a ring, a A and morphisms preserve the distinguished elements). The forgetful functor G:Ring* Ringhas a left adjoint - it assigns to every ring R the pair (R[x],x) where R[x] is the polynomial ring with coecientsfrom R.Abelianization. Consider the inclusion functor G : Ab Grp from the category of abelian groups to categoryofgroups. IthasaleftadjointcalledabelianizationwhichassignstoeverygroupGthequotientgroupGab=G/[G,G].The Grothendieck group. In K-theory, the point of departure is to observe that the category of vector bundleson a topological space has a commutative monoid structure under direct sum. One may make an abeliangroup out of this monoid, the Grothendieck group, by formally adding an additive inverse for each bundle(or equivalence class). Alternatively one can observe that the functor that for each group takes the underlyingmonoid (ignoring inverses) has a left adjoint. This is a once-for-all construction, in line with the third sectiondiscussion above. That is, one can imitate the construction of negative numbers; but there is the other optionof an existence theorem. For the case of nitary algebraic structures, the existence by itself can be referred touniversal algebra, or model theory; naturally there is also a proof adapted to category theory, too.Frobeniusreciprocity in the representation theory of groups: see induced representation. This exampleforeshadowed the general theory by about half a century.TopologyA functor with a left and a right adjoint. Let G be the functor from topological spaces to sets that associatesto every topological space its underlying set (forgetting the topology, that is). G has a left adjoint F, creatingthe discrete space on a set Y, and a right adjoint H creating the trivial topology on Y.Suspensions and loop spaces Given topological spaces X and Y, the space [SX, Y] of homotopy classes ofmaps from the suspension SX of X to Y is naturally isomorphic to the space [X, Y] of homotopy classes ofmaps from X to the loop space Y of Y. This is an important fact in homotopy theory.12 CHAPTER 1. ADJOINT FUNCTORSStone-ech compactication. Let KHaus be the category of compact Hausdor spaces and G : KHaus Top be the inclusion functor to the category of topological spaces. Then G has a left adjoint F : Top KHaus,the Stoneech compactication. The unit of this adjoint pair yields a continuous map from every topologicalspace X into its Stone-ech compactication. This map is an embedding (i.e. injective, continuous and open)if and only if X is a Tychono space.Direct and inverse images of sheaves Every continuous map f : X Y between topological spaces inducesa functor f from the category of sheaves (of sets, or abelian groups, or rings...) on X to the correspondingcategory of sheaves on Y, the direct image functor. It also induces a functor f 1 from the category of sheavesof abelian groups on Y to the category of sheaves of abelian groups on X, the inverse image functor. f 1 isleft adjoint to f. Here a more subtle point is that the left adjoint for coherent sheaves will dier from that forsheaves (of sets).Soberication. The article on Stone duality describes an adjunction between the category of topological spacesand the category of sober spaces that is known as soberication. Notably, the article also contains a detaileddescription of another adjunction that prepares the way for the famous duality of sober spaces and spatiallocales, exploited in pointless topology.Category theoryA series of adjunctions. The functor 0 which assigns to a category its set of connected components is left-adjoint to the functor D which assigns to a set the discrete category on that set. Moreover, D is left-adjoint tothe object functor U which assigns to each category its set of objects, and nally U is left-adjoint to A whichassigns to each set the indiscrete category on that set.Exponential object. In a cartesian closed category the endofunctor C C given by A has a right adjointA.Categorical logicQuantication. If Yis a unary predicate expressing some property, then a suciently strong set theory mayprove the existence of the set Y = {y | Y (y)} of terms that fulll the property. A proper subset T Yand the associated injection of T into Yis characterized by a predicate T(y) = Y (y) (y) expressing astrictly more restrictive property.The role of quantiers in predicate logics is in forming propositions and also in expressing sophisticatedpredicates by closing formulas with possibly more variables. For example, consider a predicate f withtwo open variables of sort X and Y. Using a quantier to close X , we can form the set{y Y | x. f(x, y) S(x)}of all elements y of Yfor which there is an x to which it is f -related, and which itself is characterizedby the property S . Set theoretic operations like the intersection of two sets directly corresponds tothe conjunction of predicates. In categorical logic, a subeld of topos theory, quantiers are identiedwith adjoints to the pullback functor. Such a realization can be seen in analogy to the discussion ofpropositional logic using set theory but, interestingly, the general denition make for a richer range oflogics.So consider an object Yin a category with pullbacks. Any morphism f: X Yinduces a functorf: Sub(Y ) Sub(X)on the category that is the preorder of subobjects. It maps subobjects Tof Y(technically: monomor-phism classes of T Y) to the pullback X YT . If this functor has a left- or right adjoint, they arecalled f and f , respectively.[3] They both map from Sub(X) back to Sub(Y ) . Very roughly, given adomain S X to quantify a relation expressed via f over, the functor/quantier closes X in X YTand returns the thereby specied subset of Y.1.7. PROPERTIES 13Example: In Set , the category of sets and functions, the canonical subobjects are the subset (or rathertheir canonical injections). The pullback fT= X YT of an injection of a subset T into Yalong fis characterized as the largest set which knows all about f and the injection of Tinto Y. It thereforeturns out to be (in bijection with) the inverse image f1[T] X .For S X , let us gure out the left adjoinet, which is dened viaHom(fS, T) = Hom(S, fT),which here just meansfS T S f1[T]Consider f[S] T. We see S f1[f[S]] f1[T] . Conversely, If for an x S we also havex f1[T] , then clearly f(x) T . So S f1[T] implies f[S] T . We concude that left adjointto the inverse image functor f is given by the direct image. Here is a characterization of this result,which matches more the logical interpretation: The image ofS under fis the full set ofy 's, suchthat f1[{y}] S is non-empty. This works because it neglects exactly those y Ywhich are in thecomplement of f[S] . SofS= {y Y | (x f1[{y}]). x S } = f[S].Put this in analogy to our motivation {y Y | x. f(x, y) S(x)} .The right adjoint to the inverse image functor is given (without doing the computation here) byfS= {y Y | (x f1[{y}]). x S }.The subset fS of Yis characterized as the full set of y 's with the property that the inverse image of{y} with respect to fis fully contained within S . Note how the predicate determining the set is thesame as above, except that is replaced by .See also powerset.1.7 Properties1.7.1 ExistenceNot every functor G : C D admits a left adjoint. If C is a complete category, then the functors with left adjoints canbe characterized by the adjoint functor theorem of Peter J. Freyd: G has a left adjoint if and only if it is continuousand a certain smallness condition is satised: for every object Y of D there exists a family of morphismsfi : Y G(Xi)where the indices i come from a set I, not a proper class, such that every morphismh : Y G(X)can be written ash = G(t) o fifor some i in I and some morphismt : Xi X in C.An analogous statement characterizes those functors with a right adjoint.14 CHAPTER 1. ADJOINT FUNCTORS1.7.2 UniquenessIf the functor F : C D has two right adjoints G and G, then G and G are naturally isomorphic. The same is truefor left adjoints.Conversely, if Fis left adjoint to G, and G is naturally isomorphic to G then Fis also left adjoint to G. Moregenerally, if F, G, , is an adjunction (with counit-unit (,)) and : F F : G Gare natural isomorphisms then F, G, , is an adjunction where= ( ) = (1 1).Here denotes vertical composition of natural transformations, and denotes horizontal composition.1.7.3 CompositionAdjunctions can be composed in a natural fashion. Specically, if F, G, , is an adjunction between C and Dand F, G, , is an adjunction between D and E then the functorF F: C Eis left adjoint toG G: C E.More precisely, there is an adjunction between F F and G G with unit and counit given by the compositions:1EGFGFGGFFFFGGFGFG1C.This new adjunction is called the composition of the two given adjunctions.One can then form a category whose objects are all small categories and whose morphisms are adjunctions.1.7.4 Limit preservationThe most important property of adjoints is their continuity: every functor that has a left adjoint (and therefore is aright adjoint) is continuous (i.e. commutes with limits in the category theoretical sense); every functor that has a rightadjoint (and therefore is a left adjoint) is cocontinuous (i.e. commutes with colimits).Since many common constructions in mathematics are limits or colimits, this provides a wealth of information. Forexample:applying a right adjoint functor to a product of objects yields the product of the images;applying a left adjoint functor to a coproduct of objects yields the coproduct of the images;every right adjoint functor is left exact;every left adjoint functor is right exact.1.8. RELATIONSHIPS 151.7.5 AdditivityIf C and D are preadditive categories and F : C D is an additive functor with a right adjoint G : C D, then G isalso an additive functor and the hom-set bijectionsY,X: homC(FY, X) = homD(Y, GX)are, in fact, isomorphisms of abelian groups. Dually, if G is additive with a left adjoint F, then F is also additive.Moreover, if both C and D are additive categories (i.e. preadditive categories with all nite biproducts), then any pairof adjoint functors between them are automatically additive.1.8 Relationships1.8.1 Universal constructionsAs stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one foreach object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G : C D from every object of D, then G has a left adjoint.However, universal constructions are more general than adjoint functors: a universal construction is like an opti-mization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D(equivalently, every object of C).1.8.2 Equivalences of categoriesIf a functor F: CD is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence ofcategories, i.e. an adjunction whose unit and counit are isomorphisms.Every adjunction F, G, , extends an equivalence of certain subcategories. Dene C1 as the full subcategory ofC consisting of those objects X of C for which X is an isomorphism, and dene D1 as the full subcategory of Dconsisting of those objects Y of D for which Y is an isomorphism. Then F and G can be restricted to D1 and C1and yield inverse equivalences of these subcategories.In a sense, then, adjoints are generalized inverses. Note however that a right inverse of F (i.e. a functor G such thatFG is naturally isomorphic to 1D) need not be a right (or left) adjoint of F. Adjoints generalize two-sided inverses.1.8.3 MonadsEvery adjunction F, G, , gives rise to an associated monad T, , in the category D. The functorT: D Dis given by T = GF. The unit of the monad: 1D Tis just the unit of the adjunction and the multiplication transformation : T2 Tis given by = GF. Dually, the triple FG, , FG denes a comonad in C.Every monad arises from some adjunctionin fact, typically from many adjunctionsin the above fashion. Twoconstructions, called the category of EilenbergMoore algebras and the Kleisli category are two extremal solutions tothe problem of constructing an adjunction that gives rise to a given monad.16 CHAPTER 1. ADJOINT FUNCTORS1.9 References[1] arXiv.org: John C. Baez Higher-Dimensional Algebra II: 2-Hilbert Spaces.[2] William Lawvere, Adjointness in foundations, Dialectica, 1969, available here. The notation is dierent nowadays; aneasier introduction by Peter Smith in these lecture notes, which also attribute the concept to the article cited.[3] Saunders Mac Lane, Ieke Moerdijk, (1992) Sheaves in Geometry and Logic Springer-Verlag. ISBN 0-387-97710-4 Seepage 58Admek, Ji; Herrlich, Horst; Strecker, George E. (1990). Abstract and Concrete Categories. The joy of cats(PDF). John Wiley & Sons. ISBN 0-471-60922-6. Zbl 0695.18001.Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics 5(2nd ed.). Springer-Verlag. ISBN 0-387-98403-8. Zbl 0906.18001.1.10 External linksAdjunctions Seven short lectures on adjunctions.Chapter 2Amnestic functorIn the mathematical eld of category theory, an amnestic functor F : A B is a functor for which A-isomorphism is an identity whenever F is an identity.2.1 ReferencesAbstract and Concrete Categories. The Joy of Cats. Jiri Admek, Horst Herrlich, George E. Strecker.17Chapter 3Polynomial functorIn algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a techniquefor studying functors by approximating them by a sequence of simpler functors; it generalizes the sheacation of apresheaf. This sequence of approximations is formally similar to the Taylor series of a smooth function, hence theterm "calculus of functors.Many objects of central interest in algebraic topology can be seen as functors, which are dicult to analyze directly, sothe idea is to replace them with simpler functors which are suciently good approximations for certain purposes. Thecalculus of functors was developed by Thomas Goodwillie in a series of three papers in the 1990s and 2000s,[1][2][3]and has since been expanded and applied in a number of areas.3.1 ExamplesA motivational example, of central interest in geometric topology, is the functor of embeddings of one manifold Minto another manifold N, whose rst derivative in the sense of calculus of functors is the functor of immersions. Asevery embedding is an immersion, one obtains an inclusion of functors Emb(M, N) Imm(M, N) in this casethe map from a functor to an approximation is an inclusion, but in general it is simply a map.As this example illustrates, the linear approximation of a functor (on a topological space) is its sheacation, thinkingof the functor as a presheaf on the space (formally, as a functor on the category of open subsets of the space), andsheaves are the linear functors.This example was studied by Goodwillie and Michael Weiss.[4][5]3.2 DenitionHere is an analogy: with the Taylor series method from calculus, you can approximate the shape of a smooth functionf around a point x by using a sequence of increasingly accurate polynomial functions. In a similar way, with thecalculus of functors method, you can approximate the behavior of certain kind of functor F at a particular object Xby using a sequence of increasingly accurate polynomial functors.To be specic, let M be a smooth manifold and let O(M) be the category of open subspaces of Mi.e. the categorywhere the objects are the open subspaces of M, and the morphisms are inclusion maps. Let F be a contravariantfunctor from the category O(M) to the category Top of topological spaces with continuous morphisms. This kind offunctor, called a Top-valued presheaf on M, is the kind of functor you can approximate using the calculus of functorsmethod; you want to know what sort of topological space F(X) is for a particular open set XO(M), so you study thetopology of the increasingly accurate approximations F0(X), F1(X), F2(X), and so on.In the calculus of functors method, the sequence of approximations consists of (1) functors T0F, T1F, T2F, and soon, as well as (2) natural transformations k:F TkF for each integer k. These natural transforms are required to becompatible, meaning that the composition F Tk+1F TkFequals the map F TkF, and thus form a towerF Tk+1F TkF T1F T0F, and can be thought of as successive approximations, just as in aTaylor series one can progressively discard higher order terms.183.3. BRANCHES 19The approximating functors are required to be "k-excisive" such functors are called polynomial functors by analogywith Taylor polynomials which is a simplifying condition, and roughly means that they are determined by theirbehavior around k points at a time, or more formally are sheaves on the conguration space of k points in the givenspace. The dierence between the kth and (k 1) st functors is a homogeneous functor of degree k" (by analogywith homogeneous polynomials), which can be classied.For the functors TkFto be approximations to the original functor F, the resulting approximation maps F TkFmust be n-connected for some number n, meaning that the approximating functor approximates the original functorin dimension up to n"; this may not occur. Further, if one wishes to reconstruct the original functor, the resultingapproximations must be n-connected for n increasing to innity. One then calls F an analytic functor, and says thatthe Taylor tower converges to the functor, in analogy with Taylor series of an analytic function.3.3 BranchesThere are three branches of the calculus of functors, developed in the order:manifold calculus, such as embeddings,homotopy calculus, andorthogonal calculus.Homotopy calculus has seen far wider application than the other branches.3.4 HistoryThe notion of a sheaf and sheacation of a presheaf date to early category theory, and can be seen as the linear formof the calculus of functors. The quadratic form can be seen in the work of Andr Haeiger on links of spheres in1965, where he dened a metastable range in which the problem is simpler.[6] This was identied as the quadraticapproximation to the embeddings functor in Goodwillie and Weiss.3.5 References[1] T. Goodwillie, Calculus I: The rst derivative of pseudoisotopy theory, K-theory 4 (1990), 1-27.[2] T. Goodwillie, Calculus II: Analytic functors, K-theory 5 (1992), 295-332.[3] T. Goodwillie, Calculus III: Taylor series, Geom. Topol. 7 (2003), 645-711.[4] M. Weiss, Embeddings from the point of view of immersion theory, Part I, Geometry and Topology 3 (1999), 67-101.[5] T. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory, Part II, Geometry and Topology 3(1999), 103-118.[6] Haeiger, Andr, Enlacements de sphres en codimension suprieure 2Munson, Brian (2005), Syllabus for Math 283: Calculus of Functors3.6 External linksThomas GoodwillieJohn KleinMichael S. WeissChapter 4Polynomial functorIn algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a techniquefor studying functors by approximating them by a sequence of simpler functors; it generalizes the sheacation of apresheaf. This sequence of approximations is formally similar to the Taylor series of a smooth function, hence theterm "calculus of functors.Many objects of central interest in algebraic topology can be seen as functors, which are dicult to analyze directly, sothe idea is to replace them with simpler functors which are suciently good approximations for certain purposes. Thecalculus of functors was developed by Thomas Goodwillie in a series of three papers in the 1990s and 2000s,[1][2][3]and has since been expanded and applied in a number of areas.4.1 ExamplesA motivational example, of central interest in geometric topology, is the functor of embeddings of one manifold Minto another manifold N, whose rst derivative in the sense of calculus of functors is the functor of immersions. Asevery embedding is an immersion, one obtains an inclusion of functors Emb(M, N) Imm(M, N) in this casethe map from a functor to an approximation is an inclusion, but in general it is simply a map.As this example illustrates, the linear approximation of a functor (on a topological space) is its sheacation, thinkingof the functor as a presheaf on the space (formally, as a functor on the category of open subsets of the space), andsheaves are the linear functors.This example was studied by Goodwillie and Michael Weiss.[4][5]4.2 DenitionHere is an analogy: with the Taylor series method from calculus, you can approximate the shape of a smooth functionf around a point x by using a sequence of increasingly accurate polynomial functions. In a similar way, with thecalculus of functors method, you can approximate the behavior of certain kind of functor F at a particular object Xby using a sequence of increasingly accurate polynomial functors.To be specic, let M be a smooth manifold and let O(M) be the category of open subspaces of Mi.e. the categorywhere the objects are the open subspaces of M, and the morphisms are inclusion maps. Let F be a contravariantfunctor from the category O(M) to the category Top of topological spaces with continuous morphisms. This kind offunctor, called a Top-valued presheaf on M, is the kind of functor you can approximate using the calculus of functorsmethod; you want to know what sort of topological space F(X) is for a particular open set XO(M), so you study thetopology of the increasingly accurate approximations F0(X), F1(X), F2(X), and so on.In the calculus of functors method, the sequence of approximations consists of (1) functors T0F, T1F, T2F, and soon, as well as (2) natural transformations k:F TkF for each integer k. These natural transforms are required to becompatible, meaning that the composition F Tk+1F TkFequals the map F TkF, and thus form a towerF Tk+1F TkF T1F T0F, and can be thought of as successive approximations, just as in aTaylor series one can progressively discard higher order terms.204.3. BRANCHES 21The approximating functors are required to be "k-excisive" such functors are called polynomial functors by analogywith Taylor polynomials which is a simplifying condition, and roughly means that they are determined by theirbehavior around k points at a time, or more formally are sheaves on the conguration space of k points in the givenspace. The dierence between the kth and (k 1) st functors is a homogeneous functor of degree k" (by analogywith homogeneous polynomials), which can be classied.For the functors TkFto be approximations to the original functor F, the resulting approximation maps F TkFmust be n-connected for some number n, meaning that the approximating functor approximates the original functorin dimension up to n"; this may not occur. Further, if one wishes to reconstruct the original functor, the resultingapproximations must be n-connected for n increasing to innity. One then calls F an analytic functor, and says thatthe Taylor tower converges to the functor, in analogy with Taylor series of an analytic function.4.3 BranchesThere are three branches of the calculus of functors, developed in the order:manifold calculus, such as embeddings,homotopy calculus, andorthogonal calculus.Homotopy calculus has seen far wider application than the other branches.4.4 HistoryThe notion of a sheaf and sheacation of a presheaf date to early category theory, and can be seen as the linear formof the calculus of functors. The quadratic form can be seen in the work of Andr Haeiger on links of spheres in1965, where he dened a metastable range in which the problem is simpler.[6] This was identied as the quadraticapproximation to the embeddings functor in Goodwillie and Weiss.4.5 References[1] T. Goodwillie, Calculus I: The rst derivative of pseudoisotopy theory, K-theory 4 (1990), 1-27.[2] T. Goodwillie, Calculus II: Analytic functors, K-theory 5 (1992), 295-332.[3] T. Goodwillie, Calculus III: Taylor series, Geom. Topol. 7 (2003), 645-711.[4] M. Weiss, Embeddings from the point of view of immersion theory, Part I, Geometry and Topology 3 (1999), 67-101.[5] T. Goodwillie and M. Weiss, Embeddings from the point of view of immersion theory, Part II, Geometry and Topology 3(1999), 103-118.[6] Haeiger, Andr, Enlacements de sphres en codimension suprieure 2Munson, Brian (2005), Syllabus for Math 283: Calculus of Functors4.6 External linksThomas GoodwillieJohn KleinMichael S. WeissChapter 5Derived functorIn mathematics, certain functors may be derived to obtain other functors closely related to the original ones. Thisoperation, while fairly abstract, unies a number of constructions throughout mathematics.5.1 MotivationIt was noted in various quite dierent settings that a short exact sequence often gives rise to a long exact sequence.The concept of derived functors explains and claries many of these observations.Suppose we are given a covariant left exact functor F : A B between two abelian categories A and B. If 0 A B C 0 is a short exact sequence in A, then applying F yields the exact sequence 0 F(A) F(B) F(C)and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, thisquestion is ill-posed, since there are always numerous dierent ways to continue a given exact sequence to the right.But it turns out that (if A is nice enough) there is one canonical way of doing so, given by the right derived functorsof F. For every i1, there is a functor RiF: A B, and the above sequence continues like so: 0 F(A) F(B) F(C) R1F(A) R1F(B) R1F(C) R2F(A) R2F(B) ... . From this we see that F is an exact functor ifand only if R1F = 0; so in a sense the right derived functors of F measure how far F is from being exact.If the object Ain the above short exact sequence is injective, then the sequence splits. Applying any additive functor toa split sequence results in a split sequence, so in particular R1F(A) = 0. Right derived functors are zero on injectives:this is the motivation for the construction given below.5.2 Construction and rst propertiesThe crucial assumption we need to make about our abelian category A is that it has enough injectives, meaning thatfor every object A in A there exists a monomorphism A I where I is an injective object in A.The right derived functors of the covariant left-exact functor F : A B are then dened as follows. Start with anobject X of A. Because there are enough injectives, we can construct a long exact sequence of the form0 X I0 I1 I2 where the I iare all injective (this is known as an injective resolution of X). Applying the functor F to this sequence,and chopping o the rst term, we obtain the chain complex0 F(I0) F(I1) F(I2) Note: this is in general not an exact sequence anymore. But we can compute its homology at the i-th spot (the kernelof the map from F(Ii) modulo the image of the map to F(Ii)); we call the result RiF(X). Of course, various thingshave to be checked: the end result does not depend on the given injective resolution of X, and any morphism X 225.3. VARIATIONS 23Y naturally yields a morphism RiF(X) RiF(Y), so that we indeed obtain a functor. Note that left exactness meansthat 0 F(X) F(I0) F(I1) is exact, so R0F(X) = F(X), so we only get something interesting for i>0.(Technically, to produce well-dened derivatives of F, we would have to x an injective resolution for every objectof A. This choice of injective resolutions then yields functors RiF. Dierent choices of resolutions yield naturallyisomorphic functors, so in the end the choice doesn't really matter.)The above-mentioned property of turning short exact sequences into long exact sequences is a consequence of thesnake lemma. This tell us that the collection of derived functors is a -functor.If X is itself injective, then we can choose the injective resolution 0 X X 0, and we obtain that RiF(X) = 0for all i 1. In practice, this fact, together with the long exact sequence property, is often used to compute the valuesof right derived functors.An equivalent way to compute RiF(X) is the following: take an injective resolution of X as above, and let Kibe theimage of the map Ii1Ii(for i=0, dene Ii1=0), which is the same as the kernel of IiIi+1. Let i : Ii1Kibethe corresponding surjective map. Then RiF(X) is the cokernel of F(i).5.3 VariationsIf one starts with a covariant right-exact functor G, and the category A has enough projectives (i.e.for every objectA of A there exists an epimorphism P A where P is a projective object), then one can dene analogously theleft-derived functors LiG. For an object X of A we rst construct a projective resolution of the form P2 P1 P0 X 0where the Pi are projective. We apply G to this sequence, chop o the last term, and compute homology to getLiG(X). As before, L0G(X) = G(X).In this case, the long exact sequence will grow to the left rather than to the right:0 A B C 0is turned into L2G(C) L1G(A) L1G(B) L1G(C) G(A) G(B) G(C) 0Left derived functors are zero on all projective objects.One may also start with a contravariant left-exact functor F; the resulting right-derived functors are then also con-travariant. The short exact sequence0 A B C 0is turned into the long exact sequence0 F(C) F(B) F(A) R1F(C) R1F(B) R1F(A) R2F(C) These right derived functors are zero on projectives and are therefore computed via projective resolutions.5.4 ApplicationsSheaf cohomology. If X is a topological space, then the category of all sheaves of abelian groups on X is an abeliancategory with enough injectives. The functor which assigns to each such sheaf L the group L(X) of global sections is24 CHAPTER 5. DERIVED FUNCTORleft exact, and the right derived functors are the sheaf cohomology functors, usually written as H i(X,L). Slightly moregenerally: if (X, OX) is a ringed space, then the category of all sheaves of OX-modules is an abelian category withenough injectives, and we can again construct sheaf cohomology as the right derived functors of the global sectionfunctor.tale cohomology is another cohomology theory for sheaves over a scheme.Ext functors. If R is a ring, then the category of all left R-modules is an abelian category with enough injectives. IfA is a xed left R-module, then the functor Hom(A,-) is left exact, and its right derived functors are the Ext functorsExtRi(A,-).Tor functors. The category of left R-modules also has enough projectives. If A is a xed right R-module, then thetensor product with A gives a right exact covariant functor on the category of left R-modules; its left derivatives arethe Tor functors TorRi(A,-).Group cohomology. Let G be a group. A G-module M is an abelian group M together with a group action of Gon M as a group of automorphisms. This is the same as a module over the group ring ZG. The G-modules form anabelian category with enough injectives. We write MGfor the subgroup of M consisting of all elements of M thatare held xed by G. This is a left-exact functor, and its right derived functors are the group cohomology functors,typically written as Hi(G,M).5.5 NaturalityDerived functors and the long exact sequences are natural in several technical senses.First, given a commutative diagram of the form0 A1f1 B1g1 C1 0 0 A2f2 B2g2 C2 0(where the rows are exact), the two resulting long exact sequences are related by commuting squares:Second, suppose : F G is a natural transformation from the left exact functor F to the left exact functor G. Thennatural transformations Ri : RiF RiG are induced, and indeed Ribecomes a functor from the functor category ofall left exact functors from A to B to the full functor category of all functors from A to B. Furthermore, this functoris compatible with the long exact sequences in the following sense: if0AfBgC0is a short exact sequence, then a commutative diagramis induced.Both of these naturalities follow from the naturality of the sequence provided by the snake lemma.Conversely, the following characterization of derived functors holds: given a family of functors Ri: A B, satisfyingthe above, i.e. mapping short exact sequences to long exact sequences, such that for every injective object I of A,5.6. GENERALIZATION 25Ri(I)=0 for every positive i, then these functors are the right derived functors of R0.5.6 GeneralizationThe more modern (and more general) approach to derived functors uses the language of derived categories.5.7 ReferencesManin, Yuri Ivanovich; Gelfand, Sergei I. (2003), Methods of Homological Algebra, Berlin, NewYork: Springer-Verlag, ISBN 978-3-540-43583-9Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathe-matics 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR 1269324Chapter 6Diagonal functorIn category theory, for any objecta in any category C where the producta a exists, there exists the diagonalmorphisma: a a asatisfyingk a= ida for k {1, 2} ,where k is the canonical projection morphism to the k -th component. The existence of this morphism is a con-sequence of the universal property which characterizes the product (up to isomorphism). The restriction to binaryproducts here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a di-agonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namelyequality.For concrete categories, the diagonal morphism can be simply described by its action on elements x of the object a. Namely, a(x)= x, x , the ordered pair formed from x . The reason for the name is that the image of such adiagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphismR R2on the real line is given by the line which is a graph of the equation y= x . The diagonal morphism into the inniteproduct X may provide an injection into the space of sequences valued in X ; each element maps to the constantsequence at that element. However, most notions of sequence spaces have convergence restrictions which the imageof the diagonal map will fail to satisfy.In particular, the category of small categories has products, and so one nds the diagonal functor C C C givenby(a) = a, a , which maps objects as well as morphisms. This functor can be employed to give a succinctalternate description of the product of objects within the category C : a product a b is a universal arrow from toa, b . The arrow comprises the projection maps.More generally, in any functor category CJ(here Jshould be thought of as a small index category), for each objecta in C , there is a constant functor with xed object a : (a) CJ. The diagonal functor : C CJassigns toeach object of C the functor (a) , and t