Appendix A

Table of Notation

Sets and Categories

P (A) the set of the subsets of APω(A) the set of the finite subsets of AA\B the difference between sets A and B, {x ∈ A | x �∈ B}card(A) the cardinality of the set Aλ+ the last cardinal strictly greater than the ordinal λF |J the reduction of the filter F over I to a subset J ∈ FSet the category of sets as objects and functions as arrowsClass the hyper-category of classes as objects and functions (be-

tween classes) as arrowsCat the hyper-category of categories as objects and functors as

arrowsGrp the category of groups|C| the class of objects of the category C

C(A,B) the set of arrows between objects A and Bdom( f ) the domain (source) of the arrow fcod( f ) the codomain (target) of the arrow ff ;g the composition of arrows f and gCop the opposite of the category C

C∗ the 2-opposite of a 2-category C

A∼= B the objects A and B are isomorphicA×B the (direct) product of objects A and B∏i∈I Ai the (direct) product of the family of objects {Ai}i∈I

A + B the co-product (direct sum) of the objects A and B0C the initial object of the category C

352 Appendix A. Table of Notation

Lim(D) vertex of the limiting cone for the diagram DColim(D) vertex of the co-limiting co-cone for the diagram DA�B the disjoint union of sets A and BA/U comma category( f ,B) object of comma category A/U where f : A→U(B)B� the Grothendieck category determined by the indexed cate-

gory BAF the directed diagram of products F → C for a filter F over I

anda family {Ai}i∈I of objects in C

∏F Ai the filtered product of objects {Ai | i ∈ I} by the filter F overan index set I

Institutions

SigI the category of the signatures of institution ISenI the sentence functor of institution IModI the model functor of institution IM |=I

Σ ρ the Σ-model M satisfies the Σ-sentence ρ in the institution IΓ % E entailment (i.e., there exists a proof) from Γ to EPfI the proof functor of proof system IRlI the proof rule functor of system-of-proof rules IhI ,cI the hypotheses and the conclusions for natural transforma-

tions ofsystem-of-proof rules I

(co)Ins/P f Ins the category of institution/institution with proofs(co)morphisms

(co)RlSys/P f Sys the category of system-of-proof rules/proof system(co)morphisms

ϕst/ϕop/ϕrl the mapping on sort/OPERATION/relation symbols of themorphism ϕof FOL-signatures

E∗ the class of models satisfying the set of sentences EM∗ the set of sentences satisfied by the class of models M

E• the theory generated by the set of sentences E in a proof sys-tem

E |=| E ′ E and E ′ are semantically equivalent sets of sentences, i.e.,E |= E ′ and E ′ |= E

E %& E ′ E and E ′ are proof theoretic equivalent sets of sentences, i.e.,E % E ′ and E ′ % E

M≡M′ M and M′ are elementarily equivalent classes of models

353

|M| the set of the elements of the carriers of the model M(x : s) the constant/variable x has sort s−→S the set of all (higher-order) types constructed from the sorts

SτΣ the semantic topology on Σ-modelsThI the category of theories of institution IPresI the category of presentations of institution II pres the institution of the presentations of the institution I0Σ/0Σ,E the initial [Σ/(Σ,E)]-modelModpres the model functor ModI pres

of I pres

M1⊗M2 the amalgamation of models M1 and M2

ιΣ(M) : Σ→ ΣM the elementary extension of the signature Σ via the model M(ΣM,EM) the elementary diagram of the Σ-model MiΣ,M the natural isomorphism determined by the elementary dia-

gram of the Σ-model MMM the initial model 0ΣM ,EM of the elementary diagram of a

model MNh i−1

Σ,M(h) for h : M → N model homomorphismE(I ) the elementary sub-institution of I= f the kernel of homomorphism fSw/Sc class of (plain)/closed injective model homomorphisms in

FOL and PAS f class of full subalgebras in PAHr class of surjective model homomorphisms in FOL and PAHs/Hc class of strong/closed surjective homomorphisms in FOLρ1∧ρ2 the conjunction of ρ1 and ρ2

ρ1∨ρ2 the disjunction of ρ1 and ρ2

ρ1 ⇒ ρ2 the implication of ρ2 by ρ1

ρ1 ⇔ ρ2 the equivalence between ρ1 and ρ2

te= t ′ existence equation∧E the conjunction of the set of sentences E

¬ρ the negation of ρ¬E {¬ρ | ρ ∈ E}(∀χ)ρ universal quantification of Σ′-sentence ρ for χ : Σ→ Σ′ sig-

nature morphism(∃χ)ρ existential quantification of Σ′-sentence ρ for χ : Σ→ Σ′ sig-

nature morphismTF the algebra of F-termsTF(X) the F-algebra of F-terms over the set of variables XMχ the model representing the signature morphism χiχ the natural isomorphism defining a representable signature

morphism χ

354 Appendix A. Table of Notation

Mod(ψ)/Sen(ψ) the model reduct/sentence translation part of the substitutionψ

ME the model defining a basic set of sentences EM |=inj h M is with respect to hIn j(H) the class of objects/models injective with respect to all h∈Hc(ρ1, . . . ,ρn) the c-connection of sentences ρ1, . . . ,ρn

Fixed(U) the fixed points of the semantic operator UUp(M) the class of all ultraproducts of models of M

Ur the ultraradical relation on modelsUniv(Σ) the set of universal Σ-sentences in FOLExist(Σ) the set of existential Σ-sentences in FOLM[Sen0]N M∗ ∩Sen0(Σ)⊆ N∗ ∩Sen0(Σ)

MSen0−→ N there exists a Σ-model homomorphism h : M → N such that

MM[Sen0]Nh

K-Mod(Σ) the category of K-Σ-Kripke models with models fromMod(Σ),where K ∈ {T,S4,S5}

M-Sen(Σ) the set of ‘modal’ Σ-sentences over Sen(Σ)J � the Grothendieck institution of the indexed (co)institution JSP |=| SP′ equivalence of specificationsSP� ϕ renaming of specification SPϕ|SP hiding of specification SPSpecI the category of structured specifications of institution ISpec〈T ,D〉 the category of 〈T , D〉-specificationsI spec the institution of structured specifications over II spec〈T ,D〉 the institution of 〈T , D〉-specifications over I

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Index

Cat, ‘quasi-category’ of categories, 10Ins, category of institution morphisms,

39Set, category of sets, 8false, in institution, 95fol, 256Σ11 sentence, 134〈T , D〉-specification, 327¬¬-elimination, 289fol, 258, 261coIns, category of institution comorphisms,

40Łos institution, 132Łos sentence, 1320-cell, 181-cell, 182-category, 182-cell, 182-co-cone, 192-co-limit, 192-cone, 192-dimensional opposite category, 202-functor, 192-limit, 192-natural transformation, 19

abstract inclusion, 74abstract surjection, 74accessible sentence, 96adjoint comorphism of institutions, 41adjoint morphism of institutions, 41adjunction, 16

2-categorical, 19algebra, 24algebraic signature, 24

amalgamationfor signature morphism, (Φ,β),

216of models, 60

amalgamation square, 60weak, 60

anti-additive function, 157anti-monotonic function, 157approximation equation, 36arity

of operation symbol, 23of relation symbol, 23of term, 100

arrow above an arrow, 20arrow is (λ,D)-chain, 146automaton, 36

basic sentence, 108basic set of sentences, 108Birkhoff institution, 186

symmetric, 215Birkhoff proof system, 310Boolean complete institution, 92Boolean connective, 93broad subcategory, 10

cardinal, 145cardinality of sets, 145carrier set, 24cartesian arrow, 20cartesian closed category, 17cartesian functor, 21cartesian lifting, 20cartesian morphism of institutions, 255categorical diagrams, 12

Index 369

category, 7of signatures, 27of substitutions, 99

chain in category ((λ,D)-chain), 146chain in category (λ-chain), 145charter, 38closed homomorphism of FOL models,

66, 76closed inclusion system, 76, 78, 79

for specifications, 320closed morphism

of proof theoretic presentations, 300of specifications, 320of theories, 79

closure under isomorphisms, 28closure operator, 193co-complete category, 14co-cone, 12co-equalizer, 13co-limit, 12co-product, 13co-unit, of adjunction, 16co-well powered inclusion system, 171co-well-powered category, 11coherent adjoint-indexed institution, 258comma category, 11comorphism of systems of rules, 279comorphism of institutions, 40

AUT→ FOL1, 43FOEQL→HNK, 43FOL→ (FOL1)pres, 52FOL→ FOEQL, 41FOL→HNK, 43FOL→ RELpres, 52LA→ FOEQLpres, 56MBA→ FOL, 43MPL→ REL1, 42PA→ FOLpres, operational, 53PA→ FOLpres, relational, 53PA→MApres, 56POA→ FOLpres, 55WPL→ PLpres, 56

comorphism of institutions with pre-quan-tifiers, 292

comorphism of proof systems, 279with pre-quantifiers, 292

comorphism of sentence systems withpre-quantifiers, 292

compact institution, 134compact proof, 284compact proof system, 284complete category, 14complete institution with proofs, 283complete lifting of filtered products by

functor, 129cone, 12congruence, 77congruence (Γ-congruence), 86congruence in POA, 81conjunction

in institutions, 92proof-theoretic, 288semantic, 92

connected category, 8connection of sentences, 93connective in institution, 93conservative homomorphism of models,

112conservative morphism of signatures,

64, 95consistent set of sentences, 134

proof theoretically, 296constant (operation symbol), 23constraint query, 347constraint substitution, 348contraction algebra, 36countably incomplete ultrafilter, 157Craig interpolation square,

proof-theoretic, 296Craig (L,R )-interpolation, 190Craig S -left interpolation for

comorphism, 215Craig S -right interpolation for

comorphism, 217Craig interpolation square, 190

of comorphisms, 267Craig-Robinson interpolation square,

211

370 Index

creation of (co-)limits, 14

dense morphism of signatures, 137derivation, of specification, 319derived operation, 100derived relation, 100derived signature, 100diagonal functor, 11diagrams, 12directed co-limit, 14directed model amalgamation, 61directed-exact institution, 62disjunction in institution, 92disjunction, semantic, 92distinguished cartesian morphism, 21

elementarily equivalent models, 50elementary amalgamation square, 211elementary class of models, 50elementary diagram

in FOL, 65in institution, 67

elementary embedding in FOL, 66elementary extension, 67elementary homomorphism of models,

68D-elementary, 115

elementary institution, 68elementary sub-institution, 118enriched indexed inclusion system, 266entailment, 277

institution, 298relation, 277system, 277system of presentations, 301

epi, 8epi basic sentence, 108epi basic set of sentences, 108epic inclusion system, 75epimorphic family of arrows, 8equalizer, 13equation, 25equivalence

in institution, 92

of categories, 17of institutions, 41, 257of specifications, 319of substitutions, 99

equivalence relation, S-sorted, 76equivalence, semantic, 92equivalent sentences, semantically, 50exact comorphism of institutions, 63,

262exact institution, 62

weakly, 62exact morphism

of institutions, 63of signatures, (ΦΔ,βΔ), 248

existential quantification in institution,94

existential sentence in FOL, 164expansion of models in institution, 28explicitly defined morphism of

signatures, 224proof theoretically, 297

extension, Sen0, 164

faithful functor, 10fibration, 20fibre category, 20fibre of institution, 255fibred category, 20fibred institution, 254filter, 121filtered power of models, in institution,

123filtered product of models

in FOL, 122in institution, 123

final functor, 14final object, 11finitary basic set of sentences, 108finitary Horn sentence, 110finitary morphism of signatures, 95finitary proof, 284finitary rule, 284finitary sentence, 59, 153finite model, 97

Index 371

finitely elementary class of models, 168finitely presented object, 15finitely presented theory, 50finitely sized model, 137first order quantification, 102first order variable, 97fixed point for semantic operator, 194free object, 16full functor, 10functor, 9

Galois connection, 17Generalized Continuum Hypothesis, 145Grothendieck category, 20Grothendieck construction, 20Grothendieck institution, 256Grothendieck object, 20group, 8

Herbrand model, 338Heyting algebra, 33higher order model, 35homomorphism

of preordered algebras, 33of FOL models, 24of HOL models, 35of contraction algebras, 36of Heyting algebras, 73of Kripke models, 32, 237of models in institution, 27of partial algebras, 31

homomorphisms of multialgebras, 34Horn institution, 302Horn sentence

in FOL, 30, 82in institution, 110

image of arrow, 74implication

in institution, 92proof-theoretic, 290semantic, 92

implicitly defined morphism ofsignatures, 224

proof theoretically, 297import of specifications, 320inclusion of specifications, 320inclusion system, 74inclusive functor, 75inclusive institution, 79indexed category, 20indexed comorphism-based institution,

257indexed institution, 256inductive co-limit, 14inductive model amalgamation, 61inductive-exact institution, 62infinitary proof system, 276initial object, 11injective object, 9institution, 27

(Π∪Σ)0n, 30

CatEQL (categorical equationallogic), 37

Cat+EQL (categorical equationallogic with binaryco-products), 73

QE(PA), 31QE1(PA), 31QE2(PA), 31AFOL (first order logic atoms), 71FOL+ (positive first order logic),

29FOLS (S-sorted first order logic),

44FOL∞,ω, FOLα,ω (infinitary

logics), 30HCL∞, 30HNKλ (HNK with λ-abstraction),

37HOLλ (HOL with λ-abstraction),

37SOL (second order logic), 30SOL∞,ω, 214AUT (automata), 36CA (contraction algebras), 36EQL (equational logic), 30

372 Index

FOEQL (first order equationallogic), 30

FOL (first order logic), 23HCL (Horn clause logic), 30HOL (higher order logic), 35HPOA (Horn preorder algebra), 34IPL (intuitionistic propositional

logic), 33LA (linear algebra), 37MA (multialgebras), 34MBA (membership algebra), 34MFOL (modal first order logic), 31PA (partial algebra), 30PL (propositional logic), 29POA (preordered algebra), 33REL (relational logic), 30∀∨ (universal disjunction of atoms),

187FOL1 (single-sorted first order logic),

29UNIV (universal sentences), 29REL1 (single-sorted REL), 42

institutionhaving finitary sentences, 153having saturated models, 147of presentations, 51of substitutions, 101with pre-quantifiers, 292with proof rules, 282with proofs, 282with representable substitutions, 106

interpolant, 190semantic, 193

interpretation of term in FOL model, 26intersection of signatures, 318inverse image functor, 21invertible enriched indexed inclusion

system, 266isomorphism, 8isomorphism classes, 8

Keisler-Shelah institution, 160kernel

of POA homomorphism, 81

of arrow, 13of model homomorphism, 77

Kripke modelin MFOL, 32in institution, 236

lax co-cone, 19lax co-limit, 19lax cone, 19lax limit, 19lax natural transformation, 19left adjoint functor, 16liberal institution, 84

comorphism, 86morphism, 86

liberal theory morphism, 84lifting

of filtered products by functor, 128of (co-)limits, 14of isomorphisms

by β, 242by spans, 206by squares, 206

of relationby signature morphism, weakly,

230by signature morphisms, 197

limit, 12limit ordinal, 145locally (semi-)exact indexed

coinstitution, 262locally co-complete indexed category,

260locally liberal indexed institution, 266locally presentable category, 15logical connective, 93logical kernel of signature morphism,

213

m-compact institution, 134maximally consistent set of sentences,

136modal preservation of sentence by fil-

tered product, 245

Index 373

modal preservation of sentences by fil-tered factor, 245

modal sentence functor, 240model amalgamation

for comorphisms, 63weak, 63

for institution, 60directed, 61

model compact institution, 134model functor, 27model

in FOL, 24in institution, 27

model of specification, 318model realizes finitely set of sentences,

146model realizes set of sentences, 146model transformation, 39model with predefined type, 332modification between institution

morphisms, 39modularization square, 300Modus Ponens for Sen0

for entailment systems, 309for proof systems, 310

mono, 8monoid, 8monomorphic family of arrows, 8morphism

of CA signatures, 36of FOL signatures, 25of HOL signatures, 35of charters, 44of institutions, 39

FOL→ EQL, 39FOL→MA, 43FOL→REL, 42MBA→ FOL, 43PA→ FOL, 42with elementary diagrams, 69

of presentations, 50, 298of proof systems, 279of rooms, 45

of signatures,(∗e∗), 191(b ∗ ∗), 191(i∗ ∗), 191(ie∗), 191(iei), 191(iii), 191(s∗ ∗), 191

of specifications, 319of systems of rules, 279of theories, 50

proof theoretic, 298

natural transformation, 10negation

in institutions, 92proof-theoretic, 288semantic, 92

normal elementary diagram, 116normal form of specification, 324

object above an object, 20object in category, 7opposite category, 11ordinal, 145overloading of symbols, 24

partial algebra, 31persistent adjunction, 17persistently free object, 17persistently free specification, 319persistently liberal institution

comorphism, 86persistently liberal institution

morphism, 86polynomial, (Σ,A), 346pre-institution, 38pre-quantifiers, 291precise comorphism, 226precise morphism of signatures, 226predefined model, 332preorder atoms, 33preordered algebra, 33

374 Index

presentationin institution, 50in proof systems, 298

preservationof filtered products by functor, 125of F -filtered products by functor,

125of (co-)limits, 14of logical connectives, 93of sentence

by ultrafactor, 130by directed co-limits of models,

142by filtered factor, 130by filtered product, 130by limit of models, 113by model extensions, 165by submodels, 165by ultraproduct, 130

product, 13projective object, 9projectively representable morphism of

signatures, 129proof rules, 278proof system, 276

with pre-quantifiers, 292protecting presentation morphism, 345pullback, 13pushout, 13

quantification in institution(D-quantification), 94

quasi-compact institution, 190quasi-existence equation in PA, 31quasi-representable morphism of signa-

tures, 102quasi-variety, 171query, 338quotient homomorphism, 77quotient

of model, 77of object, 171

reachable object, 170

reductof FOL model homomorphisms,

25of FOL models, 25of models in institution, 28

reduction of filter, 123refinement of specifications, 320reflection

of (co-)limits, 14of sentence by directed co-limits of

models, 142relational atom, 25representable morphism of signatures,

104representation

of institutions, 87of signature morphism, 104

retract, 8right adjoint functor, 16Robinson consistency square, 204room, 45

satisfaction conditionfor institution comorphisms, 40for institution morphisms, 39for institutions, 28

satisfaction of sentence at possibleworld, 239

satisfaction relationin FOL, 26in institution, 27

satisfaction with predefined type, 333saturated model, 146second order substitution, 101semantic entailment system, 277semantic operator, 193semantic proof system, 277semantic topology, 55semi-exact indexed coinstitution, 263semi-exact institution, 62sentence functor, 27sentence in FOL, 25sentence system with pre-quantifiers,

291

Index 375

sentence transformation, 39signature functor, 39signature

in CA, 36in FOL, 23in HOL, 35in MBA, 34in PA, 30of specification, 318

signature transformation, 39signature with predefined type, 332simple elementary diagrams, 153size of model, 153skeleton of category, 17small co-limit, 14small limit, 14small morphism of signatures, 146solution for query, 339solution form for query, 340sort of operation symbol, 23sound institution

with proof rules, 282with proofs, 282

span, 13specification, 318split fibration, 21split fibred institution, 255stability

under pushouts, 15under pullbacks, 15

stable sentence functor, 157strong FOL homomorphism, 71strong homomorphism of FOL models,

76strong inclusion system, 76, 78, 79

for specifications, 320strong morphism

of proof theoretic presentations, 300of specifications, 320of theories, 54, 79

strongly persistent adjunction, 17strongly persistently free object, 17sub-institution, 29subcategory, 10

submodel, 76generated by set, 80Sen0, 164

subobject, 170substitution in institution

D-substitution, 99Σ-substitution, 98

substitution, first order, 97substitutivity rule, 304successor ordinal, 145supporting co-limits by indexed

co-institutions, 260

theory in institution, 50theory in proof system, 298theory morphism, proof theoretic, 298tight morphism of signatures, 226translation

of FOL sentences, 26of specification, 318

translations of solutions of queries, 345triangular equations, 16

2-categorical, 19type in HOL, 35

ultrafilter, 121ultrapower of models, in institution, 123ultraproduct of models, in institution,

123ultraradical, 169unifier, 341

most general, 341union of signatures, 318union of specifications, 318unions in inclusion system, 75unit of adjunction, 16universal arrow, 11universal institution, 302universal proof system, 304universal quantification in institution, 94universal sentence in FOL, 29

variable, 25variety, 171

376 Index

vertical arrow, 20vertical composition, 10

weak amalgamation square, 60weak model amalgamation for comor-

phisms, 63weakly exact institution, 62weakly lifting of relation by signature

morphism, 230well-powered category, 11