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  • MOTIVIC INFINITE LOOP SPACES

    ELDEN ELMANTO, MARC HOYOIS, ADEEL A. KHAN, VLADIMIR SOSNILO,AND MARIA YAKERSON

    In Memory of Vladimir Voevodsky

    Abstract. We prove a recognition principle for motivic infinite P1-loop spaces over a perfect

    field of characteristic 6= 2. This is achieved by developing a theory of framed motivic spaces,which is a motivic analogue of the theory of E-spaces. A framed motivic space is a motivicspace equipped with transfers along finite syntomic morphisms with trivialized cotangent

    complex in K-theory. Our main result is that grouplike framed motivic spaces are equivalent

    to the full subcategory of motivic spectra generated under colimits by suspension spectra.As a consequence, we deduce some representability results for suspension spectra of smooth

    varieties, and in particular for the motivic sphere spectrum, in terms of Hilbert schemes ofpoints in affine spaces.

    Contents

    1. Introduction 21.1. The recognition principle in ordinary homotopy theory 21.2. The motivic recognition principle 31.3. Framed correspondences 51.4. Outline of the paper 81.5. Conventions and notation 81.6. Acknowledgments 92. Notions of framed correspondences 92.1. Equationally framed correspondences 102.2. Normally framed correspondences 142.3. Tangentially framed correspondences 203. The recognition principle 273.1. The S1-recognition principle 273.2. Framed motivic spaces 303.3. Framed motivic spectra 343.4. The GarkushaPanin theorems 353.5. The recognition principle 374. The -category of framed correspondences 414.1. -Categories of labeled correspondences 414.2. The labeling functor for tangential framings 504.3. The symmetric monoidal structure 585. Applications 625.1. Representability of the motivic sphere spectrum 625.2. Motivic bar constructions 65

    Date: June 19, 2018.E.E. and A.K. were supported by Institut Mittag-Leffler postdoctoral fellowships.M.H. was partially supported by NSF grants DMS-1508096 and DMS-1761718.M.Y. was supported by SFB/TR 45 Periods, moduli spaces and arithmetic of algebraic varieties.

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  • 2 E. ELMANTO, M. HOYOIS, A. A. KHAN, V. SOSNILO, AND M. YAKERSON

    5.3. From framed correspondences to Voevodskys finite correspondences 675.4. From the Burnside category to framed correspondences 69Appendix A. Voevodskys Lemma 70A.1. Voevodskys Lemma 70A.2. Closed gluing 72Appendix B. Finite fields 73B.1. Framed correspondences induced by monic polynomials 73B.2. Conservativity results 74References 75

    1. Introduction

    This paper answers the following question in motivic homotopy theory: what sort of algebraicstructure characterizes an infinite P1-loop space? Our answer is: transfers along framed finitesyntomic morphisms. Before we elaborate on what we mean by this, let us briefly review theanalogous question in classical homotopy theory.

    1.1. The recognition principle in ordinary homotopy theory.

    1.1.1. Recall that a spectrum is a pointed space equipped with infinitely many successivedeloopings. More precisely, it is sequence of pointed spaces (X0,X1,X2, . . . ) with equivalencesXi ' Xi+1, where X = Hom(S1,X) is the loop space of X.

    If E = (X0,X1, . . . ) is a spectrum, its underlying space X0 = E has a very rich structure.

    Being a loop space, it admits a multiplication m : X0 X0 X0 which is associative and hasinverses up to homotopy. Being a double loop space, it has a second multiplication, which mustagree with the first one and be commutative up to homotopy by the EckmannHilton argument.Being an infinite loop space means that X0 has infinitely many associative multiplications thatcommute with one another: it is an E-space.

    The recognition principle for spectra states that, if E is connective, i.e., if each space Xiis (i 1)-connected, then this additional structure on X0 determines the spectrum E up tohomotopy. Several approaches were devised to make this precise. The first statement and proofof the recognition principle are due to Boardman and Vogt [BV68]. The name recognitionprinciple was coined by May, who proved a version of it using his notion of operad [May72].

    The point of view that we shall adopt is that of Segal [Seg74]. Segal introduces the category, which is the opposite of the category Fin of finite pointed sets. Alternatively, one can thinkof as the category whose objects are finite sets and whose morphisms are spans X Y Zwhere Y Z is injective. The Segal maps are the injective maps {i} {1, . . . , n} in thiscategory. The category is now the standard way to define an E-object in any -categoryC with finite products [Lur17a, Definition 2.4.2.1]: it is a functor X: op C such that theSegal maps induce an equivalence X({1, . . . , n}) '

    ni=1 X({i}) for any n.

    Segals recognition principle is then the following equivalence of -categories [Seg74, Propo-sition 3.4]:

    (1.1.2) Spt>0 ' PShSeg()gp.

    The left hand side is the -category of connective spectra, and the right-hand side is the -category of grouplike E-spaces, i.e., presheaves of spaces on that satisfy Segals conditionand such that the induced monoid structure has inverse up to homotopy.

  • MOTIVIC INFINITE LOOP SPACES 3

    1.1.3. There are many variants of Segals theorem. Let us mention one with a more geometricflavor, involving the category Mfd of smooth manifolds. Every space X gives rise to a presheafhX : Mfd

    op Spc sending M to the mapping space Maps(M,X), which can be viewed as a(very) generalized cohomology theory for smooth manifolds. The presheaf hX has two specialproperties: it is local (i.e., it is a sheaf) and it is R-homotopy invariant. Using the fact thatsmooth manifolds are locally contractible, it is easy to see that the construction X 7 hX definesan equivalence of -categories

    (1.1.4) Spc ' PShloc,R(Mfd).

    If X = E, then the cohomology theory hX acquires additional features. For instance, iff : M N is a finite covering map between smooth manifolds, there is transfer map

    f : hX(M) hX(N)

    induced by Atiyah duality. This enhanced functoriality makes hX into a presheaf on a 2-categoryCorrfin(Mfd), whose objects are smooth manifolds and whose morphisms are correspondencesM N P, where M N is a finite covering map. It turns out that for a sheaf F on Mfd,an extension of F to Corrfin(Mfd) is just another way of encoding an E-structure on F (see[BH18, C.1]). From this point of view, Segals equivalence (1.1.2) becomes an equivalence

    (1.1.5) Spt>0 ' PShloc,R(Corrfin(Mfd))gp.

    In other words, connective spectra are equivalent to homotopy invariant sheaves on smoothmanifolds with transfers along finite covering maps, with the additional condition that themonoid structure induced by these transfers is grouplike. One can also show that the smashproduct of connective spectra is induced by the Cartesian product of smooth manifolds via Dayconvolution.

    Remark 1.1.6. The fact that transfers along finite covering maps characterize infinite loop spaceswas conjectured by Quillen before being disproved by Kraines and Lada [KL79]. This is becauseQuillen did not have the language to express the higher coherences necessary for the validity ofthe result, and therefore did not demand them.

    1.2. The motivic recognition principle.

    1.2.1. In [Mor99, MV99], Morel and Voevodsky construct the -category H(S) of motivicspaces over a base scheme S, as an algebro-geometric analog of the classical homotopy theoryof spaces. A motivic space over S is by definition a presheaf of spaces on the category SmSof smooth S-schemes that is local with respect to the Nisnevich topology and A1-homotopyinvariant:

    H(S) = PShnis,A1(SmS).

    This definition can be compared with (1.1.4). The -category SH(S) of motivic spectra overS is a certain stabilization of H(S) with respect to the pointed projective line P1, introducedby Voevodsky [Voe98]. His construction imitates that of the-category of spectra in topology:a motivic spectrum is a sequence of motivic spaces (X0,X1,X2, . . . ) with equivalences Xi 'Hom(P

    1,Xi+1) expressing Xi+1 as a P1-delooping of Xi.

    1.2.2. Our main theorem is a direct analog of (1.1.5) in motivic homotopy theory. We construct

    an -category Corrfr(SmS) of framed correspondences between smooth S-schemes, and weprove the following result:

    Theorem 1.2.3 (Motivic Recognition Principle, Theorem 3.5.13). Let k be a perfect field ofcharacteristic 6= 2. There is an equivalence of symmetric monoidal -categories

    SHveff(k) ' PShnis,A1(Corrfr(Smk))gp.

  • 4 E. ELMANTO, M. HOYOIS, A. A. KHAN, V. SOSNILO, AND M. YAKERSON

    Here, SHveff(k) is the-category of very effective motivic spectra [S12, Section 5], which isthe appropriate analog of connective spectra in motivic homotopy theory: it is the subcategoryof SH(k) generated under colimits and extensions by P1-suspension spectra. To obtain thestable -category generated under colimits by P1-suspension spectra, i.e., the -category ofeffective motivic spectra [Voe02, Section 2], it suffices to replace presheaves of spaces withpresheaves of spectra:

    SHeff(k) ' PShnis,A1(Corrfr(Smk),Spt).We will discuss the -category Corrfr(Smk) in more details below.

    1.2.4. The problem of formulating a recognition principle for infinite P1-loop spaces was firstposed by Voevodsky: in his collection of open problems in stable motivic homotopy theory[Voe02, Section 8], he mentions the hypothetical theory of operadic description of P1-loopspaces. While many of the conjectures in loc. cit. have been proved by other means, such atheory has been elusive. In fact, an operadic approach to this problem remains so.

    Voevodsky himself took important steps towards a recognition principle in several unpub-lished n

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