HOMOTOPOI

TIBOR BEKE

Abstract. This is a survey and elaboration of two aspects of homotopy in topos theory, which may be termed the gros and petit cases; more precisely, of their interplay.

Contents

Introduction 1 1. From homological to homotopical 2 2. Calculus of zig-zags and fractions 4 3. Homotopy in TOPOI 10 4. Homotopy of spaces vs. homotopy of topoi 14 5. Open doors 16 References 20

Introduction

Consider some category of objects (thought of as geometric entities) whose structure one wishes to investigate via sheaf theory. In the gros case, one endows the category with a suitable Grothendieck topology; so objects are replaced by the (sheafified) functors they represent and morphisms by natural transformations. Any category of sheaves is very wellbehaved, but it may fail to reflect the underlying geometry adequately. The issue of representability, in the guise of descent problems, becomes central.

In the petit case, one associates, functorially, a site to each object. Objects are replaced by the corresponding categories of sheaves; morphisms, by the functors induced by maps of sites. Very often, these data reflect the geometric situation well, but they are not easy to work with. The categorical data acquire an inevitable second dimension: two objects (i.e. categories of sheaves) are almost never isomorphic, merely equivalent; diagrams fail to commute on the nose, only up to a (hopefully canonical) isomorphism; even the simplest (co)limits — e.g. pullbacks — satisfy only a weaker universality property… Grothendieck’s extensive vocabulary of pseudofunctors, fibered (and cofibered) categories, etc. was invented just to handle the combinatorics.

This distinction of petit and gros may seem unclear and even arbitrary in the case of, say, the étale topology; due to special properties of étale maps, to give the étale topology on the category of schemes is tantamount to endowing each scheme, compatibly, with the étale topology, and vice versa. SGA4 describes various petit and gros topoi associated to individual schemes (spaces etc.) and proves their cohomology isomorphic. Our usage of these two terms differs from that of SGA4 (which is one of size) and of Lawvere [26] (which is one of certain qualitative differences of topoi). The intuition adopted here is, quite simply, that in the gros case a category of structured (simplicial, R-module, group…) objects of a fixed topos serves to model a homotopy theory; while in the petit case, a topos in its entirety is thought of as a geometric object, geometric topos morphisms replacing continuous maps — so one attempts “to do homotopy theory in the 2-category TOPOI”.

Date: October 16, 2000.

A link between the two approaches is provided by the theory of classifying topoi, itself a descendant (albeit by a long chain of associations!) of the homotopical problem of classifying bundles over a space. Essentially, any method of representing global homotopy classes in a sufficiently combinatorial fashion (a precise definition of which would be a diversion here) translates into “toposophic Brown representability”, Thm. 3.5. These results are generalizations of the existence of Eilenberg–MacLane topoi, that is, the representability of abelian cohomology classes in a homotopy category attached to TOPOI. Wraith’s [37] conjecture on the cohomology of Eilenberg–MacLane spaces, proved in Joyal–Wraith [25], generalizes accordingly; see section 4.

Acknowledgements. My indebtedness to various works of Artin, Coste, Giraud, Illusie, Joyal, Mazur, Moerdijk, Verdier and Wraith (in addition to the obvious name, Grothendieck) should be very clear from the text. Research on “homotopoi” was active for a good decade, and peaked in the early 80’s with the article of Joyal–Wraith [25]; the present paper is in fact a product of the desire to revive their research. I also benefited from conversations with Ieke Moerdijk, Jaap van Oosten, Gavin Wraith and Phil Hirschhorn.

Caveat lector. These notes are not meant to be published in the immediate future. One goal is to give a concrete and accessible (if condensed) exposition of topics that are often still regarded arcana, and another is to stimulate research in this field. Questions are raised freely throughout.

Terminology and notation. Topos will mean Grothendieck topos, and topos map means geometric morphism. Fix a universe V, a model of ZFC, whose elements are called simply sets; every topos will be a V -topos, though auxiliary universes may be used to form categories such as TOPOI (topoi and topos maps). For readability, we will often use the same symbol for a set and for the corresponding constant object in a topos (“constant sheaf on a site”). 1 will stand for the “singleton”, the terminal object in a category of sheaves.

Post-SGA4 breakthroughs, notably classifying topoi and the logical aspect of topos theory are employed where needed. Coherent logic is to be finitary, while geometric logic is to allow arbitrary disjunctions. For background on these matters, I will refer to MacLane-Moerdijk [27], the loving detail of whose exposition one cannot hope to match.

Running examples. Throughout the paper, the following stock examples of topoi will be used to illustrate abstract definitions, and bring them “down to earth”.

(a) Set, the category of sets and functions

(b) Pre(C), the category of presheaves on a (small) category C

(c) G-Set, G-equivariant sets and functions, for a topological group G (d) Sh(X), sheaves on a topological space X (in the usual sense of topology of open subsets)

1. From homological to homotopical

Historical remarks. Following the discovery of the basic notions of abelian categories, derived functors and simplicial homotopy theory, the decisive moments were Quillen’s formulation [31], [32] of axioms for abstract homotopy theory, and Joyal’s proof [23] that the category of simplicial objects in a Grothendieck topos satisfies these. We review the essential definitions and theorems.

Local (or “internal”) fibrations. Let ℰ^{Δ^{op}} be the category of simplicial objects in a topos ℰ. The terms global and local fibration will be used as in Jardine [21]. As applies to fibrations and Kan complexes, we will use internal interchangeably with local, since a simplicial map is a local fibration in ℰ iff it satisfies the horn-filling conditions in the sense of the internal logic of ℰ.

Local (or “internal”) weak equivalences. The (“global” simplicial) homotopy category of ℰ, denoted ho_{ℰ^{Δ^{op}}} , is by definition ℰ^{Δ^{op}}[W^{−1}], the category of simplicial objects in ℰ with the class of local weak equivalences inverted. (This is the device introduced by Gabriel– Zisman [17] to compare, in the case ℰ = Set, the simplicial and topological homotopy theories.) ho_{ℰ^{Δ^{op}}} has the same objects as ℰ^{Δ^{op}}; we will not use the equivalent description, due to Quillen, as the category of globally fibrant-cofibrant objects modulo homotopy. The localization functor ℰ^{Δ^{op}} → ho_{ℰ^{Δ^{op}}} will be denoted ho as well, and analogously for other homotopy categories.

Theorem 1.1. (Joyal) The following three classes of morphisms of ℰ^{Δ^{op}} satisfy Quillen’s axioms for a closed homotopy model category: cofibrations=monomorphisms, weak equivalences=local weak equivalences, fibrations=global fibrations. This homotopy model structure is simplicial and proper.

This theorem dates from the early 80’s (Joyal [23]). Seemingly the only way to prove it is to reduce the “local” properties to the case of ℰ = Set, and use somewhat involved transfinite combinatorics for the global factorization axioms. See Jardine [21], [22] for careful expositions of these. [6], [3] show just how automatic the reduction to Set can be made for other homotopy model categories too.

Remark 1.2. Of course, the category of models does not itself determine the homotopy theory, but there can be little doubt about the appropriate notion of weak equivalence between simplicial objects; this is due, among other reasons, to the fact that they are describable in a logically very simple fashion (by countably many countably long axioms of geometric logic); equivalently, they can be detected on stalks. The case is (co)fibrations is more subtle; there are infinitely many possibilities here, some functorial in TOPOI and some not. (It appears, nonetheless, that the only choice that makes all objects cofibrant is cofibrations=monomorphisms.) All the choices lead to the same Quillen equivalence type of a model category; see [3] for an elaboration.

Global homotopy groups. For an abelian group object A in ℰ, let K(A, n) be a preferred simplicial model of an Eilenberg–MacLane space, such as the de-normalization of A[n], the abelian sheaf A thought of as a chain complex concentrated in degree n. The three important features of K(A, n) are: being functorial in A; commuting with inverse image parts of geometric morphisms; and being locally a Kan complex.

The basic identity is (n ⩾ 0, D being the derived category)

R^{n}Γ(A) ≅ Ext_{Ab(ℰ)}^{n}(ℤ, A) ≅ D(ℤ, A[n]) ≅ ho_{Ab(ℰ)^{Δ^{op}}} (ℤ, K(A, n)) ≅ ho_{ℰ^{Δ^{op}}} (1, K(A, n))

Any of these may be chosen as the definition of abelian topos cohomology, H^{n}(ℰ, A). Let us introduce the shorthand ho(ℰ, X) for ho_{ℰ^{Δ^{op}}} (1, X), “homotopy of ℰ with simplicial coefficient X”, whose properties will be our main interest. See section 5 for natural occurrences of these sets (or groups) where X is not an Eilenberg–MacLane space.

Functoriality. ℰ ↦ ℰ^{Δ^{op}} extends to a functor TOPOI → HOMODEL where the objects of HOMODEL are Quillen model categories and its morphisms, Quillen adjoint pairs. Since the inverse image of a geometric morphism ℰ ⟶^{f} ℱ preserves all weak equivalences, its total left derived functor

\xymatrix{ℰ^{Δ^{op}} \ar[d]^{ho} & ℱ^{Δ^{op}} \ar[l]^{f^{∗}} \ar[d]^{ho} \\ _{ho_{ℰ^{Δ^{op}}}} & _{ho_{ℱ^{Δ^{op}}}} \ar[l]^{Lf^{∗}}}

exists by virtue of the universal property of ho, and makes the diagram commutative. For Rf_{∗} one of course needs fibrant replacement. For a composable pair of morphisms ℰ ⟶^{f} ℱ ⟶^{g} 𝒢, R(f · g)_{∗} ≅ Rf_{∗} · Rg_{∗}; the analogous statement for the derived category is a version of the Leray spectral sequence.

Localizing at a simplicial object. U ∈ ℰ^{Δ^{op}} induces a topos morphism ℰ^{Δ^{op}}/U → ℰ^{Δ^{op}}. The inverse image part, U^{∗}, takes X to X ×U ⟶^{pr_{2}} U, and has a left adjoint U_{!} that sends X → U to X. This is a Quillen pair, thus inducing a bijection ho_{ℰ^{Δ^{op}}/U} (1, U^{∗}X) ≅ ho_{ℰ^{Δ^{op}}} (U, X). Setting X to be an Eilenberg–MacLane simplicial object K(A, n) in ℰ, one obtains the wellknown H^{n}(ℰ^{Δ^{op}}/U, U^{∗}X) = H^{n}(ℰ, U, A), the second expression being the cohomology of U in the topos ℰ. Note that ℰ^{Δ^{op}}/U is not equivalent to the category of simplicial objects in some topos in general, though it’s the same as (ℰ/U_{0})^{Δ^{op}} if U is the simplicial object constant on U_{0} ∈ ℰ.

To go further, one needs more explicit and combinatorical access to global homotopy groups than that contained in Quillen’s classical theory.

2. Calculus of zig-zags and fractions

Let M be a category with a distinguished subcategory W, which we will assume to contain all identity morphisms of M. Maps in W will be referred to as weak equivalences. Dwyer– Kan [14], [15] provide a substantial generalization of Gabriel–Zisman theory of localizations of categories. For brevity, write ho_{M} for M[W^{−1}], the category obtained from M by formally inverting the morphisms of W. A key aspect of the work of Dwyer and Kan is viewing ho_{M} as a category enriched over SSet via “homotopical function complexes”. We will ignore here (for the time being) the higher-dimensional simplices, concentrating on the dimension 0 part, i.e. on the morphisms of ho_{M}. We begin with Dwyer–Kan’s symmetrized version of fractions.

Remark 2.1. In what follows, we work freely with categories that are not locally small (i.e. that may possess a proper class of arrows between two objects) since both localized categories and categories of fractions may be such. This does not affect the desired conclusion, which is an equivalence between two (locally not necessarily small) categories. It does require that we work freely with classes, as in Gödel–Bernays set theory. Nonetheless, the methods developed here often allow one to conclude that a localized category is locally small without proving it’s part of a homotopy model structure.

Definition 2.2. For objects U, X of M, let zig(U, X) be the category whose objects are zig-zags of length 3, U ⟵^{w_{1}} C ⟶^{f} F ⟵^{w_{2}} X where w_{1}, w_{2} are weak equivalences; C, F, f are arbitrary. Let a morphism from U ← C_{1} → F_{1} ← X to U ← C_{2} → F_{2} ← X be maps C_{1} → C_{2}, F_{1} → F_{2} that make

\xymatrix{ & C_{1} \ar[r] \ar[dl] \ar[dd] & F_{1} \ar[dd] \\ U &  &  & X \ar@{-->}[ul] \ar[dl] \\  & C_{2} \ar[r] \ar@{-->}[ul] & F_{2}}

commute.

Write π_{0}C for the class of connected components of a category C (equivalently, the connected components of the nerve of C). The localization functor M ⟶^{ho} ho_{M} induces a map (of classes, in general) π_{0}zig(U, X) ⟶^{γ} ho_{M}(U, X) by sending a representative

U ⟵^{w_{1}} C ⟶^{f} F ⟵^{w_{2}} X of a connected component to U ⟶^{(how_{1})^{−1}} C ⟶^{hof} F ⟶^{(how_{2})^{−1}} X;

this is well-defined, since if the left-hand diagram commutes, so does the one on the right:

[figure: C_{1}; c; f_{1}; w_{1}; F_{1}; f; C_{1}; hof_{1}; hoc; F_{1}; (how_{2})^{−1}; hof; U; X; w_{2}; u_{2}; γ; U; (how_{1})^{−1}; (hou_{1})^{−1}; X; C_{2}; f_{2}; u_{1}; F_{2}; C_{2}; hof_{2}; F_{2}; (hou_{2})^{−1}]

Theorem 2.3. (Dwyer–Kan) Let M, W form part of a Quillen model category. The mapping π_{0}zig(U, X) ⟶^{γ} ho_{M}(U, X) is a bijection (of sets) for any U, X ∈ M.

For a Quillen model category with functorial factorizations of maps into an (acyclic) cofibration followed by an acyclic fibration (resp. fibration), the proof is sketched in Dwyer– Kan [14], and that is all that’s needed in this paper. For the general case, see Dwyer–Kan [15].

Remark 2.4. It is useful to think of a short zig-zag as if it were a map from a cofibrant approximation to U to a fibrant approximation to X, thus an approximation to a class in ho(U, X) in Quillen’s sense. The remarkable feature of this case of the “hammock localization” is that no explicit preferred class of (co)fibrations enters, though the existence of some such is needed in the proof — to define composition of 3-term zig-zags, in fact, whence it is easy to prove that zig has the requisite universal property.

Let frac_{W^{−1}M}(U, X) be the full subcategory of zig(U, X) with objects zig-zags of the form U ⟵^{w} C ⟶^{f} X ⟵^{id_{X}} X (“right fractions”); dually, frac_{MW^{−1}}(U, X) is the subcategory of zig(U, X) spanned by zig-zags of the form U ⟵^{id_{U}} U ⟶^{f} F ⟵^{w} X (“left fractions”). The above γ restricts to maps π_{0}frac_{W^{−1}M}(U, X) → ho_{M}(U, X) resp. π_{0}frac_{MW^{−1}}(U, X) → ho_{M}(U, X), still denoted by the same letter. We seek sufficient conditions on M, W for these maps to be bijections. The following definition bears the same relation to the Gabriel– Zisman calculus of fractions as γ does to the Gabriel–Zisman map from equivalence classes of fractions to the localized category. We only spell out one of the dual cases.

Definition 2.5. Let F := U ⟵^{w} C ⟶^{f} X be a right fraction. The corresponding category of amalgamations, amal_{F}, has as objects all commutative diagrams

\xymatrix{C \ar[r]^{f} \ar[d]_{w} & X \ar[d]^{u} \\ U \ar[r]_{g} & F}

with u ∈ W, and a morphism between two such is an arrow F_{1} ⟶^{h} F_{2} making

[figure: C; f; w; X; u_{1}; u_{2}; U; g_{1}; g_{2}; F_{1}; h; F_{2}]

commute. Now say that M, W possesses a weak calculus of left fractions if for any right fraction F, amal_{F} is non-empty and connected.

Example 2.6. The Gabriel–Zisman axioms for a left calculus of fractions state that amal_{F}

is non-empty, and for every commutative diagram in M of the form • ⟶^{u} • ⇉_{g}^{f} • with

u ∈ W, there exists v ∈ W making • ⇉_{g}^{f} • ⟶^{v} • commute. But this implies that every amal_{F}

is connected (and filtered, in fact).

Example 2.7. If the class W is closed under pushout in M, then each amal_{F} is connected (in fact, possesses an initial object). Such M, W are typical examples of weak but not Gabriel-Zisman calculi.

Proposition 2.8. If M, W possesses a weak calculus of left fractions, then the localization map from the connected components of left fractions, π_{0}frac_{MW^{−1}}(U, X) ⟶^{γ} ho_{M}(U, X), is a bijection.

Proof. We will explicitly define a category W^{−1}M with the above hom-sets, together with a functor M → W^{−1}M, and prove it has the universal property required of ho_{M}. Let W^{−1}M have the same objects as M, and let the hom-set W^{−1}M(U, X) be π_{0}frac_{MW^{−1}}(U, X). To compose morphisms U ⇝ X and X ⇝ Y, select representatives of the components and fill in the inner diamond below (possible by the assumption that amal is non-empty):

\xymatrix{ &  & F \\  & F_{1} \ar@{.>}[ur] &  & F_{2} \ar@{.>}[ul]_{∼} \\ U \ar@{-->}[ur] &  & X \ar@{-->}[ul]_{∼} \ar@{-->}[ur] &  & Y \ar@{-->}[ul]_{∼}}

The composition is the connected component of the leg-wise composed U → F ← Y. This obviously depends only on the component of F in amal_{F_{1}⟵^{∼}X→F_{2}}, which is unique, and an easy verification shows that it only depends on the connected component of the representatives U → F_{1} ⟵^{∼} X, X → F_{2} ⟵^{∼} Y. A similar unicity argument shows associativity of composition. Units are the (components of) X ⟶^{id_{X}} X ⟵^{id_{X}} X. The fraction U ⟶^{w} X ⟵^{id_{X}} X, w ∈ W, has X ⟶^{id_{X}} X ⟵^{w} U as two-sided inverse. Define a functor M ⟶^{L} W^{−1}M that’s the identity on objects, and sends the arrow U ⟶^{f} X to U ⟶^{f} X ⟵^{id_{X}} X. Now any functor M ⟶^{G} C that sends weak equivalences to isomorphisms factors through L, by mapping the fraction U ⟶^{f} F ⟵^{w} X to G(w)^{−1}·G(f) in C (this descends through connected components, just like in the case of two-sided fractions). Moreover, this factorization is unique, since F ⟶^{id_{F}} F ⟵^{w} X, being the inverse of X ⟶^{w} F ⟵^{id_{F}} F, must perforce be mapped to G(w)^{−1}, and any morphism in W^{−1}M is a composition of one of the form U ⟶^{f} F ⟵^{id_{F}} F with an F ⟶^{id_{F}} F ⟵^{w} X. Setting M ⟶^{G} C to be the universal localization M ⟶^{ho} ho_{M}, one obtains the conclusion.

By simple inspection of the colimit of a (possibly, large!) functor into Set, one obtains

Corollary 2.9. Let X/W be the full subcategory of X/M containing as objects the maps X ⟶^{w} F with w ∈ W. Write hom(U, −) for the functor X/W → Set that sends X ⟶^{w} F to hom_{M}(U, F). If M, W possess a weak calculus of left fractions, then colim_{X/W} hom(U, −) bijects with ho_{M}(U, X).

Unlike in the case of a Gabriel–Zisman calculus of left fractions, however, it is neither true that the indexing category X/W is necessarily filtered, nor that the localization map preserves (and creates) finite colimits.

Now the notion of calculus introduced above is not the weakest possible; one could also consider an up-to-homotopy version, where the squares (amalgams) are only required to commute in a category gotten from C by quotienting out an equivalence relation (“elementary homotopies”) on the hom-sets. (This presumes that cylinder or path objects exist, so there’s more of a homotopy model structure on M than the bare subcategory of weak equivalences.) In fact, π_{0}frac_{MW^{−1}}(U, X) ⟶^{γ} ho_{M}(U, X) can be a bijection for all objects U, X without (M, W) satisfying a weak calculus of left fractions. (This is the case for (ℰ, W); see Prop. 2.15. I am indebted to Phil Hirschhorn for an explicit example of a right fraction F in (SSet, W) such that amal_{F} is empty.)

Remark 2.10. Dwyer–Kan [14] do introduce a condition based on (M, W) alone, that of a homotopy calculus of fractions that ensures that γ is a bijection. While that criterion is not easy to check, they state that it’s sufficient that W satisfy the 2-of-3 property, and functorial completions of fractions to commutative squares exist; or that W be closed under pushout in W. (The conclusion they obtain is also stronger, namely a weak equivalence between homotopy function complexes.) The reason for our discussion of weak calculus of fractions was to give direct and simple proofs of what we need.

Proposition 2.11. Let M be a Quillen model category, and AcCof the subcategory of acyclic cofibrations. Then π_{0}frac_{M AcCof^{−1}}(U, X) ⟶^{γ} ho_{M}(U, X) is a bijection if both U and X are cofibrant.

Proof. AcCof is closed under pushout in M, and it is well-known that the map M_{C}[AcCof^{−1}] → M[W^{−1}] induced by the inclusion (M_{C}, AcCof) ↪ (M, W) is an equivalence (where W is the class of weak equivalences, and M_{C} the full subcategory of M consisting of cofibrant objects).

Representing global simplicial homotopy classes. Specialize now to the case when M is ℰ^{Δ^{op}} for a topos ℰ, and W is the class of local weak equivalences. Quillen’s axioms are selfdual, but it is far from the case that the homotopy model structure on ℰ^{Δ^{op}} given by Thm. 1.1 is symmetric. Every object is cofibrant, and the class of acyclic cofibrations is definable in coherent logic (a fortiori, is of local nature) while neither of these holds for global fibrations. (Indeed, we will obtain expressions for global homotopy classes with distinct “local” and “global” duals.) But first, a review of some classical results that tie in well with the above.

Simplicial homotopy. Two maps X ⇉_{g}^{f} Y ∈ ℰ^{Δ^{op}} are simplicially homotopic if a map X ×

Δ[1] ⟶^{h} Y interpolates between them, where Δ[1] is the constant simplicial interval with its two endpoints. This relation is compatible with composition. Let π be the category ℰ^{Δ^{op}} quotiented by the equivalence relation generated by simplicial homotopy on the hom-sets.

Hypercovers. A hypercover of an object U ∈ ℰ^{Δ^{op}} is a map in π with target U that possesses a representative in ℰ^{Δ^{op}} that is an acyclic local fibration; a hypercover of a topos ℰ is (by convention) a hypercover of the terminal object of ℰ^{Δ^{op}}. Let hyp/U be the full subcategory of π/U with objects the hypercovers of U. Let α be the map (of classes, a priori)

(2.1)  colim_{C∈(hyp/U)^{op}}π(C, X) ⟶^{α} ho_{ℰ^{Δ^{op}}} (U, X)

that sends a representative U ⟵^{q} C ⟶^{f} X of a simplicial homotopy class to (hoq)^{−1} · (hof). (Simplicially homotopic maps belong to the same global homotopy class.)

Theorem 2.12. (i) (Verdier) α is a bijection for U = 1, X = K(A, n). (ii) (K. Brown–Jardine) α is a bijection if U and X are internally Kan. Historical remarks. Part (i) is SGA4, Tome 2, Exposé V, Appendice 7. (Verdier ascribes the idea of hypercovers to P. Cartier.) The version of (ii) due to Ken Brown [7] derives the statement from the axiomatic viewpoint of a “category of fibrant objects”. The axioms result in a definition of π slightly different from simplicial homotopy in general (though the two coincide in the present case). He works with sheaves on a topological space, and takes a hypercover to be a map in π possessing a representative that is any weak equivalence. The formula in full generality appears in Jardine [20], [21].

One way or another, the proof rests on two facts. Denote by ℰ_{F}^{Δ^{op}} the full subcategory of ℰ^{Δ^{op}} whose objects are locally Kan, and analogously for π_{F}.

(1) hypercovers satisfy a Gabriel–Zisman calculus of right fractions in π_{F} (a fortiori hyp/U, when restricted to π_{F}, is cofiltered)

(2) writing LAFib for the class of local acyclic fibrations, the functor ℰ_{F}^{Δ^{op}}[LAFib^{−1}] → ho_{ℰ^{Δ^{op}}} induced by the inclusion (ℰ_{F}^{Δ^{op}}, LAFib) ↪ (ℰ^{Δ^{op}}, W), is an equivalence.

Remark 2.13. hyp/U, which is a priori cofiltered but large, does possess a final small subcategory. In effect, this observation (which tends to be overlooked) is one way to prove that ho_{ℰ^{Δ^{op}}} is locally small.

Now the homotopy model-theoretic dual of Cor. 2.11 would employ global fibrations and globally fibrant objects; note however that the proof, in the case of (ℰ^{Δ^{op}}, W), continues to be valid for local fibrations and locally fibrant objects. (It needs only that local fibrations are preserved by pullbacks, and fact (2) above.) The situation can be summed up in the commutative diagram

\xymatrix{colim_{C∈(hyp/U)^{op}}π(C, X) \ar[rr]^{β} \ar@{-->}[dr]_{α} &  & π_{0}frac_{LAFib^{−1} ℰ^{Δ^{op}}} (U, X) \ar[dl]^{γ} \\  & ho_{ℰ^{Δ^{op}}} (U, X)}

(2.2)

Here α is that of (2.1), and γ is the localization map for right fractions. β is well-defined only when U, X are locally fibrant, and sends a representative U ⟵^{q} C ⟶^{f} X of a simplicial homotopy class in the colimit to its connected component in frac_{LAFib^{−1} ℰ^{Δ^{op}}} (U, X). When U, X are locally fibrant, all three of α, β and γ are bijections.

Anodyne extensions. The concept dual to hypercover is the anodyne extension of Gabriel– Zisman, that is, quite simply, acyclic cofibration, i.e. a monomorphism that’s a weak equivalence (considered as a map in π). Denote the full subcategory of X/π with objects the anodyne extensions X → F by X/anex. The diagram dual to (2.2) is

\xymatrix{colim_{F ∈X/anex}π(U, F) \ar[rr]^{β} \ar@{-->}[dr]_{α} &  & π_{0}frac_{ℰ^{Δ^{op}} AcCof^{−1}}(U, X) \ar[dl]^{γ} \\  & ho_{ℰ^{Δ^{op}}} (U, X)}

(2.3) β sends a representative U ⟶^{f} F ⟵^{a} X of a simplicial homotopy class in the colimit to its connected component in frac_{ℰ^{Δ^{op}} AcCof^{−1}}(U, X). It is well-defined for all U, X, since if f_{1}, f_{2} are homotopic, the homotopy U ⟶^{h} F^{Δ[1]} verifies the fractions U ⟶^{f_{i}} F ⟵^{a} X end up in the same connected component:

\xymatrix{ & F \\ U \ar[ur]^{f_{1}} \ar[r]^{h} \ar[dr]_{f_{2}} & F^{Δ[1]} \ar[u]_{∼} \ar[d]^{∼} & X \ar[l] \ar[ul]_{a} \ar[dl]^{a} \\  & F}

and analogously for homotopic a_{1}, a_{2}. The three maps in (2.3) are in fact bijections for all U, X.

Theme and variations. We seek to get a handle on ho_{ℰ^{Δ^{op}}} in three stages: first, by finding zig-zags as short as possible to represent homotopy classes (length 2 always suffices, but length 1 usually not); secondly, by imposing “as little as possible” algebraic data on the maps in the fractions (this is the present stage); and thirdly (in the next section) by passing to cofinal subcategories of fractions whose simplicial components are easy to describe. The most important variant specializes to (and was inspired by) a step in the Joyal–Wraith construction of Eilenberg–MacLane topoi.

We wish to replace frac_{LAFib^{−1} ℰ^{Δ^{op}}} by frac_{W^{−1}ℰ^{Δ^{op}}} in an expression for global homotopy classes. Consider the commutative diagram

\xymatrix{π_{0}frac_{LAFib^{−1} ℰ^{Δ^{op}}} (U, X) \ar[rr]^{ι} \ar[dr]_{γ_{1}} &  & π_{0}frac_{W^{−1}ℰ^{Δ^{op}}} (U, X) \ar[dl]^{γ_{2}} \\  & ho_{ℰ^{Δ^{op}}} (U, X)}

(2.4)

The γ_{i} are the localization maps, while ι is induced by the inclusion functor frac_{LAFib^{−1} ℰ^{Δ^{op}}} (U, X) ↪ frac_{W^{−1}ℰ^{Δ^{op}}} (U, X). When X is globally fibrant, ι is surjective: given U ⟵^{w} C ⟶^{f} X, factor w as an acyclic cofibration followed by a local fibration C ⟶^{j} C̃ ⟶^{p} U. By the 2-of-3 property of weak equivalences, p ∈ W. By the assump-

tion that X is globally fibrant, there exists a lift C̃ ⟶^{f̃} X such that f = f̃j. Thence

ιπ_{0}(U ⟵^{p} C̃ ⟶^{f̃} X) = π_{0}(U ⟵^{w} C ⟶^{f} X). Now if in addition U is locally fibrant, then γ_{1} is a bijection, so one has

Proposition 2.14. The localization map π_{0}frac_{W^{−1}ℰ^{Δ^{op}}} (U, X) → ho_{ℰ^{Δ^{op}}} (U, X) is a bijection when U is locally and X globally fibrant.

A dual argument establishes

Proposition 2.15. The localization map π_{0}frac_{ℰ^{Δ^{op}}W^{−1}}(U, X) → ho_{ℰ^{Δ^{op}}} (U, X) is a bijection for any U, X.

There is an important “stray” variant on Prop. 2.14. Set U to be the terminal object 1.

Proposition 2.16. The localization map π_{0}frac_{W^{−1}ℰ^{Δ^{op}}} (1, X) → ho_{ℰ^{Δ^{op}}} (1, X) =: ho(ℰ, X) is a bijection when X is locally fibrant.

Proof. It is enough to prove that the ι of (2.4) is onto. Take any 1 ⟵^{w} C ⟶^{f} X and factor f as a weak equivalence followed by a local fibration, C ⟶^{u} C̃ ⟶^{p} X. There’s a unique map C̃ → 1 and it is a weak equivalence from the 2-of-3 property. Since X was assumed to be locally Kan and p is a local fibration, C̃ is locally fibrant as well, i.e. it is a hypercover of ℰ. So the connected component of 1 ⟵^{w} C ⟶^{f} X is in the image of ι.

Question 2.17. Is there a result that unifies Props. 2.14 and 2.16? What is the largest subcategory of ℰ^{Δ^{op}} (or more generally, of a homotopy model category) whose localization can be obtained as connected components of a category of fractions?

One can establish easy analogues of (2.14)-(2.16) for fractions of type frac_{W^{−1}fib}(U, X), fib being the class of local fibrations (this is mentioned towards the end of Wraith [37] as well, at least when U = 1), or frac_{W^{−1}fib} when fib is the class of global fibrations, both of which cases have as dual frac_{cofW^{−1}}, cof being the class of monomorphisms. (One can in fact show that frac_{cofW^{−1}} satisfies a weak calculus of left fractions — suitably amended to cover the case when numerators are restricted to a subcategory of M — in any “left proper” homotopy model category, i.e. where the pushout of a weak equivalence along a cofibration is a weak equivalence.) But as fractions, all of these are more complicated than MW^{−1}.

3. Homotopy in TOPOI

Historical remarks. Yoneda’s Ext theory exhibits the set underlying Ext^{n}(A, B) as connected components — or “zig-zag equivalence classes” — of the category of models in the underlying abelian category of a certain finite limit theory. (Yoneda also describes the additive structure on the Ext-groups and their composition product, aspects that we will ignore for a little while.) In particular,

H^{n}(ℰ, A) ≅ π_{0} Mod_{𝔹Ext(ℤ,A)}(ℰ) (3.1)

Here A is an abelian group in Set, ℰ a topos, and the left-hand side is to be understood in Grothendieck’s sense. 𝔹Ext(ℤ, A) denotes the theory of extensions of abelian groups of length n between the constant groups ℤ and A; and Mod_{T} (ℰ) is the category of T -models in ℰ. This bijection, first observed in the early 70’s, is one of the earliest indicators of the interplay between the logical and topological aspects of a Grothendieck topos.

Since TOPOI has products, one can use any topos ℰ equipped with two (topos-theoretic)

points Set ⇉_{p_{1}}^{p_{0}} ℰ to define a notion of “naive homotopy”. (There is no reason a priori for

the relation thus introduced on the classes of geometric morphisms between topoi to be symmetric or transitive. But it is reflexive and preserved by composition.) The first toposinterval to come to mind is probably Sh([0, 1]), sheaves on the topological unit interval, the points being induced by 0, 1 ∈ [0, 1]. This notion is extensively investigated in Moerdijk [30].

Or consider the simplicial interval Δ[1]. Its “topos model” can be taken to be SSet/Δ[1] or Set^{Elts(Δ[1])} (which is contravariant resp. covariant Set-functors on the category of elements of Δ[1]), equipped with the topos-theoretic points coming from the two zero-simplices Δ[0] ⇉ Δ[1], and this gives a version of “simplicial topos homotopy”. But the notion that is intrinsic to the 2-categorical geometry of TOPOI, and ties in directly with its logical aspect, is the

following: an elementary homotopy between two geometric morphisms ℰ ⇉_{g}^{f} ℱ is a natural

transformation f^{∗} → g^{∗} (equivalently, f_{∗} → g_{∗}); and natural homotopy is the equivalence relation generated by elementary homotopies on the hom-classes in TOPOI. The phrase natural homotopy, and its terminological derivatives, were introduced by Joyal–Wraith [25].

Notation 3.1. For topoi ℰ, ℱ, write [ℰ, ℱ] for π_{0} Mod_{ℱ}(ℰ) = π_{0} GeoMor(ℰ, ℱ).

[ℰ, ℱ] permits other interpretations. Consider Mor(Set), the category of arrows of Set; it is equipped with two (topos-theoretic) points whose inverse images are the domain and target functors Mor(Set) → Set. (Of course, Mor(Set) is presheaves on the category with a unique non-identity arrow, • → ⋆, as well as the category of sheaves on the Sierpinski space {∘•}, the sole non-trivial topological space with two points.) [−, −] corresponds to Mor(Set) as “interval”. There are geometric morphisms Mor(Set) → Sh([0, 1]) and Mor(Set) → SSet/Δ[1] that preserve the endpoints, so the notion of natural homotopy is (strictly, as it is easy to see) finer than that generated by the topological or simplicial intervals. In fact, for

any twice-pointed topos Set ⇉_{p_{1}}^{p_{0}} ℰ, to give a geometric morphism Mor(Set) → ℰ that preserves

the two distinguished points precisely amounts to giving an elementary homotopy from p_{0} to p_{1}. Natural homotopy may therefore be thought of as the “finest” or “infinitesimal” homotopy that is at least an equivalence relation.

Proposition 3.2. [ℰ, ℱ] is a set.

Proof. The category of geometric morphisms from ℰ to ℱ, Mod_{ℱ}(ℰ), is equivalent to the category of models of a small sketch in ℰ. It is therefore accessible (cf. Makkai–Paré [28]), with a set G of dense generators. As every object of Mod_{ℱ}(ℰ) is a colimit of a diagram in G, the cardinality of the set of components of Mod_{ℱ}(ℰ) is at most that of G.

Recall that typically there exists a proper class of topos maps from ℰ to ℱ; so it’s somewhat surprising (but reassuring) that quotienting with “infinitesimal” homotopy already collapses it to a set.

Definition 3.3. For a (discrete) abelian group A and n ∈ ℕ, K(A, n) is an Eilenberg– MacLane topos if [ℰ, K(A, n)] naturally bijects with H^{n}(ℰ, A) for every topos ℰ.

As indicated above, such K(A, n) exist and are well-defined (by the Yoneda lemma) up to natural homotopy equivalence. In their paper [25], Joyal and Wraith exhibited several other Eilenberg–MacLane topoi. Recalling the identification of cohomology groups with global simplicial homotopy classes, one of their results can be stated as the existence of a bijection, compatible with (3.1),

ho(ℰ, K(A, n)) ≅ [ℰ, 𝔹(ac/K(A, n))] (3.2) for a constant abelian group A. Here 𝔹(ac/K(A, n)) is the classifying topos of the following countably axiomatizable geometric theory: a model is C → K(A, n), a simplicial object C equipped with a map to the constant K(A, n) such that C is acyclic, i.e. the unique C → 1 is a weak homotopy equivalence.

Unlike (3.1), formula (3.2) makes no explicit appeal to underlying structures of homological algebra, and it becomes natural to guess that

ho(ℰ, X) ≅ [ℰ, 𝔹(ac/X)] (3.3)

for a bigger class of simplicial sets X; here the theory ac/X is gotten by replacing K(A, n) by X in the definition of ac/K(A, n). This may be thought of as a “non-abelian”, that is to say, homotopical analogue of (3.1) or as a version of Brown representability for topoi. Indeed, we already have enough information to establish (3.3) when X is a Kan complex in SSet (which makes the corresponding constant X locally Kan, i.e. locally fibrant, in ℰ^{Δ^{op}}). Here’s the careful statement; setting X = K(A, n) then recovers the result of Joyal–Wraith.

Proposition 3.4. Fix a Kan complex X ∈ SSet. Consider the contravariant functor TOPOI ⟶^{h} Set that takes ℰ to ho(ℰ, Γ^{∗}X) where Γ is the canonical topos map ℰ → Set (with direct image given by global sections). h is representable via natural homotopy classes by 𝔹(ac/X), the classifying topos of acyclic simplicial objects over X:

ho(ℰ, Γ^{∗}X) ≅ [ℰ, 𝔹(ac/X)]

This is just Prop. 2.16 for Γ^{∗}X ∈ ℰ^{Δ^{op}}; the functoriality is immediate.

Prop. 3.4 permits generalizations in two directions. First, the source 1 in ho_{ℰ^{Δ^{op}}} (1, Γ^{∗}X) =: ho(ℰ, X) can be replaced by any constant object; and secondly, representability is not special to the homotopy theory of simplicial sets. [3], [4] introduced a type of Quillen model categories, christened sheafifiable, that extend functorially from structures based in Set to sheaves thereof. Simplicial sets is the motivating example, and there are many more:

– simplicial universal algebras (e.g. groupoids, groups, rings…)

– categories, with categorical equivalences or with nerve-equivalences as weak equivalences

– groupoids, hypergroupoids, crossed complexes…

– cosimplicial simplicial sets, bisimplicial sets, cyclic sets (with the fine or the coarse weak equivalences)

– G-equivariant simplicial sets, G a finite group (any of the fine model structures)

– homological localizations of simplicial sets

– diagram categories in any of the aforegoing. They share the following two properties: the category of models (of homotopy types) is Mod_{S}(ℰ), the category of models (in the logical sense) of a finite limit structure S in the topos ℰ; and the subcategory of weak equivalences is definable by geometric axioms in the language of morphisms of S-structures. (Their class of cofibrations is typically not axiomatizable in geometric logic, nor is it uniquely determined, but as Quillen model categories they are always cofibrantly generated, yielding, in particular, functorial factorizations of maps.) This meshes beautifully with the hammock localization theory of Dwyer–Kan.

Theorem 3.5. Let S be a sheafifiable homotopy model category as above. Fix two objects U, X ∈ Mod_{S}(Set). Consider the contravariant functor TOPOI ⟶^{h} Set that takes ℰ to ho_{Mod_{S}(ℰ)}(Γ^{∗}U, Γ^{∗}X) where Γ is the canonical topos map ℰ → Set. h is representable via natural homotopy classes by 𝔹(zig(U, X)), the classifying topos of three-term zig-zags between (the constants) U, X:

ho_{Mod_{S}(ℰ)}(Γ^{∗}U, Γ^{∗}X) ≅ [ℰ, 𝔹(zig(U, X))]

The proof is by inspection of Thm. 2.3 and the definability properties of sheafifiable homotopy model theories.

One has a result of this type for any representation of global homotopy classes by connected components of a category of zig-zags, provided the species of zig-zags in question possesses a classifying topos, i.e. is the category of models of a geometric theory. For example, in the case of fractions U ⟶^{f} F ⟵^{g} X, the structure to be possessed by f, F, g must be axiomatizable in geometric logic, and that possessed by the constants U, X must be invariant under inverse images of geometric morphisms.

The representing topos, explicitly. So far in the discussion, the key step from passing an algebraic-combinatorial description of homotopy classes to representing them in TOPOI, that is, the construction (actually, functor) 𝔹(−), has been kept in a black box. There are several (necessarily equivalent) constructions of 𝔹(T) for a geometric theory T. The first one to appear in print, Makkai–Reyes [29], exhibits 𝔹(T) as the formal completion (under coproducts and coequalizers of equivalence relations) of a category Def(T) constructed out of the syntactic data for T. This topos completion is the same as passing to sheaves on Def(T) for a certain (syntactic) Grothendieck topology; see MacLane–Moerdijk [27]. But there’s a semantic approach as well, fragments of which were essential motivation for the subject: the case of a finite limit theory goes back to Gabriel–Ulmer [16], and that of certain varieties of rings to Hakim [19]; see also the end of Makkai–Reyes [29] and Tierney [35]. It can be summed up as follows. Let T be a geometric theory, and assume T has enough models in Set. Write T as an extension of a finite limit theory L in the same language. (This can always be done, in more than one way.) Write Mod_{L} for the category of Set-models of L. It is a locally finitely presentable category; let C_{L} be its full subcategory of finitely presentable objects.

Now, in any category one can formulate the property of an object A being cone-injective w.r.t. a collection {X ⟶^{f_{λ}} Y_{λ} | λ ∈ Λ} of morphisms with the same source: this is said to be the case if for any map X ⟶^{f} A, there exists at least one μ ∈ Λ and map Y_{μ} ⟶^{g} A s.t. f = g · f_{μ}. (See Adámek–Rosický [1] for more facts on this concept.) Let I be the set of cones (i.e. sets of morphisms with common source) in C_{L} w.r.t. which any Set-model of T — thought of as an object of Mod_{L} — is cone-injective. Turning arrows around, the I generate a topology J in C_{L}^{op}, and 𝔹(T) = Sh(C_{L}^{op}, J).

It was the insight of Joyal that a variant of this technique allows one to describe certain Eilenberg–MacLane topoi concretely — in terms nearly as combinatorial as the simplicial abelian group models of Eilenberg–MacLane spaces.

Theorem 3.6. (Joyal) 𝔹(ac/X) is a presheaf topos, the category of covariant Set-valued functors on fac/X, the full subcategory of SSet/X consisting of (simplicially) finite acyclic sets over X.

There are two key ingredients to Joyal’s [25] proof. The geometric one is that any acyclic simplicial set is a filtered colimit of acyclic simplicial sets that are simplicially finite (i.e. have only finitely many non-degenerate simplices; equivalently, are finitely presentable objects of SSet). The other one is a model-theoretic criterion for deciding when a theory has a presheaf classifying topos. It is useful in many other situations (having nothing to do with homotopy theory), some of which are explored in [5].

Any topos ℰ that classifies (via natural homotopy) ho(−, Γ^{∗}X) must carry a tautologous class in ho(ℰ, Γ^{∗}X), corresponding to the identy ℰ → ℰ. For 𝔹(ac/X) = Set^{fac/X}, the tautologous class is represented by the fraction 1 ← A ⟶^{f} Γ^{∗}X where f is the natural transformation that sends A ⟶^{f} X ∈ fac/X to A ⟶^{f} X.

By the Yoneda lemma, any two topoi classifying the same functor TOPOI → Set (via natural homotopy) are naturally homotopy equivalent. When it comes to the zig-zag “theme and variations”, several such comparison maps can be made explicit, since they result simply from specializing theories. E.g. both 𝔹(frac_{W^{−1}ℰ^{Δ^{op}}} (U, X)) and 𝔹(frac_{ℰ^{Δ^{op}}W^{−1}}(U, X)) are subtopoi of 𝔹(zig(U, X)).

Working over a base. It’s not essential to Prop. 3.4 that the ground topos is Set. For non-constant coefficients, one has

Proposition 3.7. Fix a base topos ℱ and an internal Kan complex X ∈ ℱ^{Δ^{op}}. Consider the contravariant functor TOPOI/ℱ ⟶^{h} Set that takes ℰ to ho(ℰ, f^{∗}X) where f is the structure map ℰ ⟶^{f} ℱ. h is representable via natural homotopy classes in TOPOI/ℱ by 𝔹_{ℱ}(ac/X), the classifying ℱ-topos of acyclic simplicial objects over (the ℱ-constant) X:

ho(ℰ, f^{∗}X) ≅ π_{0}(TOPOI/ℱ(ℰ, 𝔹_{ℱ}(ac/X)))

This relative classifying topos allows a fairly simple construction as a pullback

\xymatrix{𝔹_{ℱ}(ac/X) \ar[r] \ar[d] & 𝔹(ac/Δ^{op}) \ar[d]^{t} \\ ℱ \ar[r]^{x} & 𝔹(Δ^{op})}

(3.4)

where 𝔹(Δ^{op}) is the classifying topos for simplicial objects, x classifies X ∈ ℱ^{Δ^{op}}, 𝔹(ac/Δ^{op}) is the classifying topos for simplicial maps with an acyclic source, and t classifies the target of the universal map.

There’s an analogous relative version of 3.5.

4. Homotopy of spaces vs. homotopy of topoi

Background. Consider the category Top of compactly generated topological spaces, SSet of simplicial sets and Cat of small categories. Each is equipped with a class of weak equivalences whose inversion results in what is usually called “the homotopy category of spaces”. They are connected by functors that induce equivalences of homotopy categories. The ones of interest to us here are the singular complex and geometric realization (whose properties will not be reviewed), the nerve functor Cat ⟶^{N} SSet, and the category of simplices: recall that for a simplicial set X, this is its category of elements as a presheaf, i.e. the comma category 𝒮_{X} := Δ↓X, here Delta (by abuse of notation) standing for the Yoneda functor Δ → SSet. This functor SSet → Cat is a sort of “homotopy adjoint” to N; more precisely, there are natural weak equivalences N𝒮_{X} → X and 𝒮_{NC} → C.

In addition, each of these three categories of models is equipped with a particularly natural functor into TOPOI:

(4.1)  Top : T ↦ Sh(T)
(4.2)  SSet : X ↦ Set^{Δ↓X}
(4.3)  Cat : C ↦ Set^{C}

(See Remark 4.1 as to why the “particularly natural” functor SSet → TOPOI is not X ↦ SSet/X, and Cat → TOPOI is not C ↦ Pre(C)!) As regards (4.1), we will further restrict to locally connected spaces that possess a basis of contractible opens, e.g. CW-complexes. Then each of the three functors (4.1)–(4.3) takes values within locally connected topoi, and each has the property that it takes morphisms that are weak equivalences in the respective sense — that would be weak homotopy equivalences for topological spaces, Kan’s combinatorial weak equivalences for simplicial sets, and functors inducing a weak equivalence between the nerves in the case of small categories — to weak equivalences between topoi in the sense of Artin–Mazur [2]. Recall that Artin and Mazur associate a pro-object in the homotopy category of simplicial sets to any locally connected topos, and define a topos morphism to be a weak equivalence iff it induces isomorphism on homotopy pro-groups. There’s a reformulation involving π_{0}, π_{1} and cohomology with locally constant coefficients; see also Wraith [36], Moerdijk [30]. The topoi associated via the functors (4.1)–(4.3) to a space resp. simplicial set resp. small category will have isomorphic cohomology and homotopy groups to those of the original object.

In what follows, weak equivalence will mean Artin–Mazur weak equivalence in the context of locally connected topoi, and retain the above listed senses in the case of spaces, simplicial sets and small categories; there should result no confusion.

There is also a comparison map “across” that only comes into existence in TOPOI, constructed by Moerdijk [30]: Sh(|NC|) → Pre(C), from sheaves on the geometric realization of a category to presheaves on it, and this map is a weak equivalence as well.

The upshot is that by taking any Eilenberg–MacLane object in Top, SSet or Cat, even regarding it first (if one wishes) as belonging to another category via one of the four functors above, then passing to the associated topos, one arrives at one and the same Artin–Mazur homotopy type. (I suspect the phenomenon extends to other models of topological homotopy equipped with “good” functors into TOPOI.)

Wraith’s conjecture. Explicit examples of Eilenberg–MacLane topoi K(A, n) with n = 0, 1 led Wraith to conjecture that they have cohomology (with constant coefficients) isomorphic to those of the Eilenberg–MacLane spaces K(A, n) for any n, the isomorphism being induced by the map Sh(K(A, n)) → K(A, n) classifying the tautologous class in H^{n}(K(A, n), A). He proved this conjecture in joint work with Joyal. (A good reference on the subject is Wraith [37]; it lists seven pertinent and deep “questions and remarks” — several of which have been motivational for this paper — in an appendix not present in the final version, Joyal–Wraith [25].) Their argument can be adapted to prove something much stronger; this is to which we turn now.

There is an inclusion i : Δ↓X ↪ fac/X, since any simplex Δ[n] ∈ SSet is weakly homotopy equivalent to the point, and has finitely many non-degenerate simplices, so is an object of fac. Apply Quillen’s [33] Theorem A to i now. For an object A→X of fac/K, the comma category i↓(A→X) is isomorphic to Δ↓A whose nerve is weakly equivalent to A, which was assumed acyclic. So the theorem applies, and i is a weak equivalence (i.e. induces a weak equivalence on the simplicial nerves).

A theorem of Cox [9] asserts that the functor Func : Cat → TOPOI takes weak equivalences of categories to weak equivalences of locally connected topoi in the sense of Artin–Mazur [2]. Weakly equivalent topoi have isomorphic (locally constant coefficient) cohomology groups. But Set^{fac/X} is an Eilenberg–MacLane topos when X is an Eilenberg–MacLane space, by Joyal’s theorem 3.6, and more generally classifies ho(−, Γ^{∗}X) when X is a fibrant simplicial set (cf. Prop. 3.4). The nerve of Δ↓X is weakly equivalent to X, and Cox also proves that the pro-homotopy type of a functor category Set^{C} is that of the nerve of C; so Set^{Δ↓X} has cohomology with constant coefficients (and homotopy) isomorphic to X. Finally, the essential geometric morphism induced by i is the map that classifies the tautologous class in ho(Set^{Δ↓X}, Γ^{∗}X). So this proves Wraith’s conjecture, for this particular model of the homotopical classifying topos.

But any two topoi classifying the simplicial homotopy type of X — meaning, topoi representing the functor ho(−, Γ^{∗}X) via natural homotopy classes — if locally connected, are weakly equivalent. Indeed, any two such representing topoi are naturally homotopy equivalent. Now there exists a continuous endpoint-preserving map from the unit interval [0, 1] to the Sierpinski space {∘•}, taking [0, 1) to the open point ∘ and 1 to the closed one, •. A fortiori, if two topos morphisms are naturally homotopic, they are homotopic via (sheaves on) the unit interval. But interval-homotopic topos maps induce isomorphic maps on cohomology with locally constant coefficients, and on pro-homotopy groups (when defined), cf. Moerdijk [30]. So naturally homotopy equivalent topoi (if locally connected) are weakly equivalent.

Now as regards models of simplicial sets resp. CW -complexes in TOPOI that result by using the singular set, topological realization, nerve and category of simplices functors, followed by taking sheaves on a topological space resp. presheaves or covariant Set-functors on a category (in any meaningful combination): as pointed out above, these all yield locally connected topoi, any two of which are Artin–Mazur equivalent. The upshot is: for a Kan complex X, any topos classifying the homotopy type of X is weakly equivalent to any of the standard models of X in TOPOI.

Remark 4.1. It is tempting to take as the “standard topos model” of X ∈ SSet the slice category SSet/X; this is in fact what Joyal–Wraith [25] do. They then observe that SSet/X = Pre(Δ↓X) = Func((Δ↓X)^{op}). But any small category C and its opposite C^{op} are weakly equivalent; in fact Quillen [33], as the first illustration of his Theorem A, exhibits a zig-zag of weak equivalences C^{op} ⟵^{s} Arr(C) ⟶^{t} C where Arr(C) is the category of elements

of the hom-bifunctor C^{op} × C ⟶^{hom(−,−)} Set, and the source resp. target maps induce s, t. At this point one can invoke Cox’s theorem again.

From the point of view of “gros homotopoi”, however, SSet/X isn’t a very natural topos model of X. In particular, it is hard to give the tautologous class in ho(SSet/X, Γ^{∗}X). Also, Cox’s theorem gives a natural identification of the nerve of C with the the pro-homotopy type of Func(C), not of Pre(C).

5. Open doors

Someday these ideas will no doubt be incorporated into a general homotopy theory for topoi. D. Quillen, on his Theorems A and B, in [33]

Proposition 5.1. For X ∈ SSet a Kan complex, there is a natural square of maps:

\xymatrix{[𝔹(ac/U), 𝔹(ac/X)] \ar[r]^{γ} & ho(𝔹(ac/U), X) \ar[d]^{i^{∗}} \\ ho_{SSet}(U, X) \ar[u]^{B} \ar[r] & ho(Set^{Δ↓U}, X)}

γ is the one of Prop. 3.4 for ℰ := 𝔹(ac/U) = Set^{fac/U}. i^{∗} is induced by i : Δ↓U ↪ fac/U from the proof of Wraith’s conjecture. B sends a representative U ⟶^{f} X of a homotopy class into (the natural homotopy class of) the essential morphism induced by composition with f: 𝔹(ac/U) = Func(fac/U) → Func(fac/X) = 𝔹(ac/X). Such a representative must exist, since X was assumed fibrant; and if f_{1}, f_{2} are simplicially homotopic, the homotopy — via the endpoint inclusions of Δ[0] in Δ[1] — induces a zig-zag of natural transformations between the two functors fac/U → fac/X, hence a zig-zag of natural homotopies between the corresponding 𝔹(ac/U) → 𝔹(ac/X). So B is well-defined. The bottom composite sends U ⟶^{f} X to the global homotopy class of the map of co-presheaves given by {Δ[n] ⟶^{σ} U} ↦ {Δ[n] ⟶^{fσ} X} on objects.

γ is a bijection by Prop. 3.4. For X an Eilenberg–MacLane simplicial set, i^{∗} is just the induced map on cohomology, thus an isomorphism by the Joyal–Wraith theorem.

Conjecture 5.2. The maps in the square are bijections for any Kan complex X.

This would imply that i^{∗} induces a bijection ho(Set^{Δ↓U}, X) ← ho(𝔹(ac/U), X) for any X ∈ SSet. (It is a “homotopy equivalence with constant coefficients”. Can the other sides be extended to arbitrary X?)

Write TOP for the full subcategory of TOPOI consisting of locally connected topoi, and ho(TOP) for TOP[W^{−1}] where W is the Artin–Mazur class of weak equivalences. Let ho(SSet) be the usual homotopy category of simplicial sets; since the functor K ↦ SSet/K preserves weak equivalences, it descends to one on the the homotopy categories.

Problem 5.3. Understand the functor ho(SSet) → ho(TOP) induced by K ↦ SSet/K. Is it full? Faithful?

5.2 and 5.3 are related as follows. Let TOP/ ∼_{S} stand for TOP modulo the equivalence relation generated on the hom-classes by natural transformation (Sierpinski homotopy). Write Kan for the full subcategory of SSet consisting of Kan complexes, and Kan/ ∼_{s} for Kan complexes modulo simplicial homotopy. One has a diagram

\xymatrix{Kan/ ∼_{s} \ar[r]^{B} \ar[d]^{i} & TOP/ ∼_{S} \ar[d] \\ ho(SSet) \ar[r] & ho(TOP)}

that commutes up to natural isomorphism, and where i is an equivalence. One can build similar diagrams with topological spaces or categories on the left-hand side, for either of which there is a notion of homotopy equivalence that, on a suitable subcategory of models, gives an equivalent description of the homotopy category. TOP carries naturally a “finest” interval, Sh({∘•}), and a faithful copy of the simplicial situation with respect to it. But there certainly exist other choices for an interval — including those coming from Top, SSet or Cat — and I suspect that TOP has enough “room” to model the geometric notion of homotopy with respect to any of them.

Problem 5.4. Find the coarsest notion of homotopy w.r.t. which the induced maps on H^{n}(−, A) or more generally on ho(−, X) are isomorphic.

Ways to compare. Let us now look more systematically at the senses in which a map ℰ ⟶^{f} ℱ may be a weak equivalence in a (hypothetical, as yet!) homotopy theory of topoi. (NH) f is a natural homotopy equivalence, defined as in Joyal–Wraith [25], w.r.t. the Sier-

pinski space as interval. (IH) f is a homotopy equivalence w.r.t. (sheaves on) the topological interval [0, 1]; this notion

is investigated extensively in Moerdijk [30]. (HO) For any X ∈ SSet, ho(ℰ, X) ← ho(ℱ, X) is a bijection. (CO) f induces isomorphisms on cohomology with constant coefficients. (W E) f is a weak equivalence, as defined — for maps between locally connected ℰ, ℱ only —

by Artin–Mazur [2], Wraith [36]. (QE) f induces a Quillen equivalence of simplicial homotopy categories ho_{ℰ^{Δ^{op}}} ⇆ ho_{ℱ^{Δ^{op}}} .

Results of this paper may be summarized as

Proposition 5.5. There are implications

\xymatrix{(NH) \ar[r] \ar[drr] & (IH) \ar[r] & (W E) \ar[r] & (CO) \\ (QE) \ar[rr] &  & (HO) \ar@{-->}[ur]}

Conjecture 5.6. None of the implications is reversible.

A candidate for (HO) ↛ (NH) is Sh(B) → Set, where B is a complete atomless Boolean algebra (with its canonical topology). As a candidate for (CO) ↛ (HO), consider SSet/X → SSet/Y where X → Y ∈ SSet induces a cohomology isomorphism without being a weak homotopy equivalence.

Variations. There’s a number of possibilities.

• For any of the above properties: demand that not only ℰ ⟶^{f} ℱ but every localized map ℰ/f^{∗}U ⟶^{f/U} ℱ/U is to possess it. (For (HO) and (CO), this works out to mean that f induces bijections ho_{ℱ^{Δ^{op}}} (f^{∗}U, f^{∗}X) ← ho_{ℰ^{Δ^{op}}} (U, X) for any object U.)

• For (HO) or (CO): iso induced with locally constant coefficients, or arbitrary coefficients from ℱ.

• Homotopy theories other than that of simplicial objects.

Question 5.7. Are there examples of maps locally (NH) or (W E) in the first sense?

(W E), as well known, implies (CO) with locally constant coefficients. The Whitehead theorem allows the reverse implication if one knows the effect on π_{0} and π_{1}.

(QE) seems very strong. I submit

Conjecture 5.8. There are no non-trivial cases of (QE). That is, if ℰ ⟶^{f} ℱ induces a Quillen equivalence on simplicial objects, then f already is an equivalence.

Question 5.9. Suppose X and Y are weakly homotopy equivalent topological spaces. What is the relation between the Quillen model structures on Sh(X)^{Δ^{op}} and Sh(Y)^{Δ^{op}}?

Comparisons of topos models. Some of the morphisms below come into existence only in TOPOI and some exist beforehand, but they deserve to be weighed on the (QE)−(NH)− (IH) − (W E) − (HO) − (CO) “scale”.

(i) for X ⟶^{f} Y a weak equivalence in SSet, compare • Func(Δ↓X) → 𝔹(ac/X)

• SSet/X → SSet/Y

• 𝔹(ac/X) → 𝔹(ac/Y) (ii) Moerdijk [30]: Sh(BC) → Pre(C)

where C is a small category, BC its (topological) classifying space. Moerdijk proves (W E).

(iii) Deligne [10], Moerdijk [30]: Sh |Y_{•}| → Sh_{•}(Y)

for a simplicial space Y_{•}; Deligne proves (CO), Moerdijk (W E) (iv) Joyal–Moerdijk [24]: cohomology isomorphism (with arbitrary coefficients) between an

arbitrary topos and a localic one

(v) Butz–Moerdijk [8]: cohomology isomorphism (with arbitrary coefficients) between an arbitrary topos with enough points and a spatial one

(vi) “petit” vs. “gros” topos, for topological, étale, Zariski etc. cases

SGA4 proves (CO), MacLane–Moerdijk (NH) (a “strong deformation retract”, in fact) (vii) Pre(−) : Cat → TOPOI sends functors to geometric morphisms and natural transfor-

mations to natural homotopies; so if C has an initial (or terminal) object, Pre(C) is naturally contractible, a fortiori (HO) with the point, Set. Suppose C ⟶^{F} D satisfies the criterion of Theorem A. Is the induced Pre(C) → Pre(D) a (HO)? Are Func(C) and Pre(C) (HO)-equivalent via a zig-zag?

H^{2}. Let G be a group object, BG its nerve; then ho(ℰ, G) resp. ho(ℰ, BG) are the usual non-abelian H^{0} and H^{1}. If C is a braided category, then H^{i}(ℰ, C) (i=0,1,2) can be defined. Compare this with Giraud’s H^{2}. Are there natural examples of braided coefficients?

Note that one could take G to be a simplicial group, and use \overline{W} G in place of BG. What is the braided analogue?

Quillen homology. Let A be e.g. a category of universal algebras, so that Quillen’s comparison theorems at the end of [31] apply. Let J be the canonical topology on A; note that Sh(A, J) is a category in the same universe as A, since the site has a set of “topological generators” in the sense of SGA4. The canonical functor A ⟶^{ϵ} Sh(A, J) has an adjoint. Abelian groups objects in both A and Sh(A, J) form an abelian category, and Quillen shows (in effect) that this adjunction induces isomorphisms on the derived category. Show that it induces a Quillen equivalence on the homotopy theories of simplicial objects.

Étale homotopy type. Dwyer–Friedlander [13] produce pro-simplicial approximations to étale topoi that ought to be looked at. Schreiderer [34] gives the homotopy type of a field (in a variant of the étale topology) as a pro-Boolean space. The granddaddy of these is, of course, the Artin–Mazur [2] comparison theorem for complex analytic spaces in the classical and étale topologies.

Cohomology operations. Just like in the topological case, ho_{ℰ^{Δ^{op}}} (K(A, n), K(B, m)) is the set of natural (not necessarily additive) cohomology operations from H^{n}(ℰ, A) to H^{m}(ℰ, B). These operations are “internal” to ℰ; more precisely, they will act on the cohomology of any topos over ℰ. The unique ℰ → Set gives a mapping

ho_{ℰ^{Δ^{op}}} (K(A, n), K(B, m)) ← ho_{SSet}(K(A, n), K(B, m))

Is it an injection for constant A, B?

Representing Steenrod operations. Joyal–Wraith prove that ho_{SSet}(K(A, n), K(B, m)) biject with the “external” operations [𝔹(ac/K(A, n))], 𝔹(ac/K(B, m))]. Find a representative of each natural homotopy class that corresponds to a finite limit definable functor.

Take now any cohomology theory and represent it by an Ω-spectrum K_{•}. If conjecture 5.2 holds, ho_{SSet}(K_{n}, K_{m}) ≅ [𝔹(ac/K_{n}), 𝔹(ac/K_{m})], then (to borrow a phrase of Joyal–Wraith) every K_{•}-operation is “expressible in certain model-theoretic terms”. It seems unlikely, though, that finite limit definable operations suffice.

Working over a base. A fundamental example is G-equivariant sheaves on X, where G is a discrete group acting on a space X. This is augmented over G-sets, so the operations etc. should come not from SSet but G-SSet (with the appropriate homotopy theory). The analogue where G is a topological (e.g. Lie) group is not understood; probably one has to work in the gros topos.

Classifying via natural homotopy. Functors of the form [−, ℱ], or “Brown representability”, deserves study of its own.

Question 5.10. Is there a Puppe sequence for [−, ℱ] or [ℰ, −]? A long exact cohomology sequence of a cofibration(?) for topos cohomology?

← … H^{n}(ℰ, A) ← H^{n}(ℱ, A) ← H^{n}(C_{f}, A) ← …

where C_{f} is the “mapping cone” of ℰ ⟶^{f} ℱ. For the case of an open/closed subtopos?

What makes (2.2) stand out of the usual pattern of classifying topos theory is that the construction represented (the formation of global homotopy classes) is highly transfinitary, and representation is via natural homotopy classes, as opposed to actual morphisms in the 2-category TOPOI.

Question 5.11. Does Joyal’s recognition theorem for theories of presheaf type relativize? Is the 𝔹_{ℱ}(ac/X) of (3.4 (internal) presheaves on ℱ?

Question 5.12. Is there a logic ℒ to “correspond to” classifying via natural homotopy? Meaning, for any theory T in ℒ, there is to exist a topos 𝔹(T) with a universal model s.t. for any topos ℰ, the set(!) of T -models in ℰ bijects with [ℰ, 𝔹(T)].

For example, any Eilenberg–MacLane topos K(A, n) carries a universal n-cohomology class such that for any topos ℰ and n-cocycle c in ℰ, there is a unique natural homotopy class ℰ → K(A, n) that pulls back the universal cocycle to c. This is the precise analogue of the original sense of “classifying space”.

In the topological context, K-groups (equivalence classes of complex, real, or complex self-conjugate etc. bundles) allow representation via homotopy classes. Is there an analogue in TOPOI?

Note that the structures being classified via natural homotopy are “second-order” in the sense that their definition entails existential quantification on types (objects, morphisms).

Question 5.13. Is the fundamental group (pointed) [S_{•}, E], where S_{•} is some “pro-circle”?

Note that the heuristic behind this question is probably wrong; what should be of interest is not the pro-discrete fundamental group but the fundamental groupoid, an object that should exist over ℰ, not just over Set.

References

[1] Jiří Adámek and Jiří Rosický. Locally presentable and accessible categories. Number 189 in London Math. Soc. Lecture Note Series. Cambridge University Press, 1994.

[2] Michael Artin and Barry Mazur. Etale homotopy. Number 100 in Lecture Notes in Mathematics. Springer-Verlag, 1969.

[3] Tibor Beke. Sheafifiable homotopy model categories. See http://www.math.uu.nl/people/beke/.

[4] Tibor Beke. Sheafifiable homotopy model categories, II. See http://www.math.uu.nl/people/beke/.

[5] Tibor Beke. Theories of presheaf type and amalgamation of models. See http://www.math.uu.nl/people/beke/.

[6] Tibor Beke. Homotopy theory and topoi. PhD thesis, Massachusetts Institute of Technology, 1998. See http://www.math.uu.nl/people/beke/.

[7] Kenneth S. Brown. Abstract homotopy theory and generalized sheaf cohomology. Transactions of the American Mathematical Society, 186:419–458, 1974.

[8] Carsten Butz and Ieke Moerdijk. Topological representation of sheaf cohomology of sites. Compositio Math., 118(2):217–233, 1999.

[9] David A. Cox. Homotopy limits and the homotopy type of functor categories. Proc. Amer. Math. Soc., 58:55–58, 1976.

[10] Pierre Deligne. Théorie de Hodge. II. Number 40 in Inst. Hautes Études Sci. Publ. Math., pages 5–57. 1971.

[11] John Duskin. Simplicial methods and the interpretation of ”triple” cohomology. Number 163 in Memoires of the AMS. American Mathematical Society, 1975.

[12] John Duskin. Higher-dimensional torsors and the cohomology of topoi: the abelian theory. In Applications of sheaves, number 753 in Lecture Notes in Mathematics, pages 255–279. Springer-Verlag, 1979.

[13] William G. Dwyer and Eric M. Friedlander. Topological models for arithmetic. Topology, 33(1):1–24, 1994.

[14] William G. Dwyer and Daniel M. Kan. Calculating simplicial localizations. Journal of Pure and Applied Algebra, 18(1):17–35, 1980.

[15] William G. Dwyer and Daniel M. Kan. Function complexes in homotopical algebra. Topology, 19(4):427– 440, 1980.

[16] Pierre Gabriel and Friedrich Ulmer. Lokal präsentierbare Kategorien, volume 221 of Lecture Notes in Mathematics. Springer-Verlag, 1971.

[17] Pierre Gabriel and Michel Zisman. Calculus of fractions and homotopy theory. Number 35 in Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1967.

[18] Paul G. Glenn. Realization of cohomology classes in arbitrary exact categories. Journal of Pure and Applied Algebra, 25(1):33–105, 1982.

[19] Monique Hakim. Topos annelés et schémas relatifs. Number 64 in Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1972.

[20] J. F. Jardine. Simplicial objects in a grothendieck topos. In Applications of algebraic K-theory to algebraic geometry and number theory, Part I, number 55 in Contemporary Mathematics, pages 193–239. American Mathematical Society, 1986.

[21] J. F. Jardine. Simplicial presheaves. Journal of Pure and Applied Algebra, 47(1):35–87, 1987.

[22] J. F. Jardine. Boolean localization, in practice. Documenta Mathematica, 1(13):245–275, 1996.

[23] André Joyal. Letter to Alexander Grothendieck. Dated 11/4/84.

[24] André Joyal and Ieke Moerdijk. Toposes are cohomologically equivalent to spaces. American Journal of Mathematics, 112(1):87–95, 1990.

[25] André Joyal and Gavin Wraith. Eilenberg-MacLane toposes and cohomology. In Mathematical applications of category theory, number 30 in Contemporary Mathematics, pages 117–131. American Mathematical Society, 1984.

[26] William F. Lawvere. Qualitative distinctions between some toposes of generalized graphs. In Categories in computer science and logic, number 92 in Contemporary Mathematics, pages 261–299. American Mathematical Society, 1989.

[27] Saunders MacLane and Ieke Moerdijk. Sheaves in Geometry and Logic. Springer-Verlag, 1992.

[28] Michael Makkai and Robert Paré. Accessible categories: the foundations of categorical model theory. Number 104 in Contemporary Mathematics. American Mathematical Society, 1989.

[29] Michael Makkai and Gonzalo E. Reyes. First order categorical logic. Model-theoretical methods in the theory of topoi and related categories. Number 611 in Lecture Notes in Mathematics. Springer-Verlag, 1977.

[30] Ieke Moerdijk. Classifying spaces and classifying topoi. Number 1616 in Lecture Notes in Mathematics. Springer-Verlag, 1995.

[31] Daniel Quillen. Homotopical algebra. Number 43 in Lecture Notes in Mathematics. Springer-Verlag, 1967.

[32] Daniel Quillen. Rational homotopy theory. Annals of Mathematics (2), 90:205–295, 1969.

[33] Daniel Quillen. Higher algebraic K-theory, I. In Algebraic K-theory, I: Higher K-theories, number 341 in Lecture Notes in Mathematics, pages 85–147. Springer-Verlag, 1973.

[34] Claus Scheiderer. Real and étale cohomology. Number 1588 in Lecture Notes in Mathematics. Springer-Verlag, 1994.

[35] Myles Tierney. Forcing topologies and classifying topoi. In Algebra, topology, and category theory, pages 211–219. Academic Press, 1976.

[36] Gavin C. Wraith. Lectures on elementary topoi. In Model theory and topoi, number 445 in Lecture Notes in Mathematics, pages 114–206. Springer-Verlag, 1975.

[37] Gavin C. Wraith. Toposes and simplicial sets: the cohomological connection. A report on joint work with A. Joyal. In Category theoretic methods in geometry, number 35 in Various Publ. Ser., pages 281–290. Aarhus University, 1983.

Mathematical Institute, Utrecht University, Utrecht, Netherlands

E-mail address: beke@math.uu.nl