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Home , End (topology), Homotopy, Homotopy theory, Nerve (category theory)

Modeling Theories

Julia E. Bergner

Throughout , each field has fundamental ob- which induce isomorphisms on all groups. In jects of study, whether it be real-valued functions or vector these examples, there is no reason to assume that such a spaces or Lie algebras or smooth . Furthermore, function has any kind of meaningful function going in the we are not only interested in what these objects are but reverse direction at all. These two situations provide clas- also in appropriate kinds of functions between them, such sical examples of “homotopy theories.” as linear transformations of vector spaces or smooth func- Our aim in this article is to give an overview of the devel- tions between manifolds. Of particular interest are those opment of the notion of , starting with functions that tell us that two specific objects are the same these classical examples and leading up to more modern as each other in some critical way. For example, we typ- perspectives. As we discuss briefly at the , we have cho- ically do not distinguish between two different sets that sen one of several possible entry points into this subject each have three elements, since they are in with and in particular do not say much about the higher categor- one another. Similarly, we ignore the difference between ical aspects of the theory. We simply remark that the “ho- groups that are isomorphic or topological spaces that are motopy theories” that we discuss here coincide with the homeomorphic. In each of these examples, the criterion “(∞, 1)-categories” that are being used widely in a num- for “sameness” is quite rigid: there is a function of the ap- ber of related fields. propriate flavor (function of sets, homomorphism, or continuous ) that admits an inverse. Topological Spaces and Chain Complexes In other situations, we have notions of “sameness” that We begin our investigations with the classical homotopy are not quite so strict. For instance, since it is generally theory of topological spaces. Let us first recall the precise extremely difficult to determine whether there is a homeo- definition of homotopy equivalence. between two topological spaces, one might ask 푓∶ 푋 → 푌 instead whether there is a homotopy equivalence between Definition 1. A continuous map of topological them, namely, a map that may not have an inverse on the spaces is a homotopy equivalence if there exists a continuous 푔∶ 푌 → 푋 푔∘푓 nose but only up to homotopy. map such that is homotopic to the identity 푋 푓 ∘ 푔 Many further examples can be described via invariants. map of and is homotopic to the identity map of 푌 One might consider weak homotopy equivalences of topo- . logical spaces, which induce isomorphisms on all homo- To define the related notion of weak homotopy equiva- topy groups, or quasi-isomorphisms of chain complexes, lence, we need to consider the homotopy groups of a topo- logical space. The most familiar of these groups is the fol- Julia Bergner is a professor of mathematics at the University of Virginia. Her lowing. email address is [email protected]. For permission to reprint this article, please contact: Definition 2. Let 푋 be a with a chosen [email protected]. basepoint 푥0. Then its is 휋1(푋, 푥0) = 1 DOI: https://doi.org/10.1090/noti1957 [푆 , 푋]∗, the set of homotopy classes of basepoint-

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1423 that the composite 푊 → ∗ → 푊 is homotopic to the identity on 푊. However, we had to use a rather exotic example of a topological space to see the difference between our two notions of equivalence, so we might ask if they agree for maps between spaces that are sufficiently nice. A good an- swer comes from the following two theorems. Whitehead’s Theorem. Any weak homotopy equivalence be- tween CW complexes is a homotopy equivalence. A CW complex is a topological space that, roughly speaking, is built by gluing together cells of different Figure 1. A quasi-circle. . Such spaces are considered well-behaved be- cause they can be described so explicitly, but the following 푆1 → 푋 preserving maps which has a natural group op- theorem also tells us that they are sufficiently general to eration. capture much of the behavior of topological spaces. While not as frequently presented in an introductory CW Approximation Theorem. Given any topological space course, higher homotopy groups can be defined analogous- 푋, there exists a weak homotopy equivalence 푌 → 푋 with 푌 a ly. CW complex. 푋 Definition 3. Let be a topological space with basepoint Taking these two theorems together, we can transition 푥 푛 ≥ 1 푛 푋 0. Then for any , the th of is from the usual of topological spaces and contin- 휋 (푋, 푥 ) = [푆푛, 푋] 푛 0 ∗. uous maps to a new category in which weak homotopy This definition also works when 푛 = 0, but in that case equivalences become isomorphisms. This category still 휋0(푋) is simply the set of components of 푋 and does has all topological spaces as objects, but given two spaces not have a natural group structure. 푋 and 푌, we define the set of maps 푋 → 푌 to consist of When 푛 ≥ 1, the group 휋푛(푋, 푥0) does not depend homotopy classes of maps between CW replacements of on the specific basepoint 푥0 but only on the path compo- 푋 and 푌. Since homotopy equivalences have inverses up nent from which it is chosen. Hence, in what follows we to homotopy, their corresponding homotopy classes have implicitly assume that appropriate statements hold for all inverses in this new category, which we call the homotopy path components and simply write the homotopy groups category of spaces. as 휋푛(푋). Now let us look at an algebraic example for which the An important feature of homotopy groups is that they same kind of phenomenon occurs. Let 푅 be a ring, and are functorial: given a continuous map of topological consider nonnegatively graded chain complexes of spaces 푓∶ 푋 → 푌, there are induced group homomor- 푅-modules phisms 푓∗ ∶ 휋푛(푋) → 휋푛(푌). 휕1 휕2 퐶• = (퐶0 ← 퐶1 ← 퐶2 ← ⋯).

Definition 4. A continuous map of topological spaces Each 퐶푖 is an 푅-module, and each composite 푋 → 푌 is a weak homotopy equivalence if, for every 푛 ≥ 0, 휕푖 휕푖+1 the map 퐶푖−1 ← 퐶푖 ← 퐶푖+1 휋푛(푋) → 휋푛(푌) is the zero map. Given another such 퐷•, a is an isomorphism. chain map 푓∶ 퐶• → 퐷• consists of 푅-module homomor- 푓 ∶ 퐶 → 퐷 푖 ≥ 0 One can check, using functoriality of homotopy groups, phisms 푖 푖 푖 for each so that the diagram o o o that any homotopy equivalence is a weak homotopy equiv- 퐶0 퐶1 퐶2 ⋯ alence. However, the converse does not hold, as the fol- 푓0 푓1 푓2 lowing example illustrates.    o o o Example 5. Let 푊 be the quasi-circle in which an arc of 퐷0 퐷1 퐷2 ⋯ 1 1 푆 is replaced by a sin( 푥 ) curve and its limit , as commutes. depicted in Figure 1. Then the unique map 푊 → ∗ is a Just as for the example of topological spaces, there are weak homotopy equivalence, since the homotopy groups two natural ways to think of equivalences between chain of both spaces are all trivial. However, this map is not a complexes. First, there is the notion of chain homotopy homotopy equivalence, since there is no map ∗ → 푊 such equivalence, which means having an inverse up to chain

1424 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 homotopy. There is also the weaker notion of quasi- following the primary example of topological spaces, are isomorphism or map that induces isomorphisms on all ho- called and , which are subject to several mology groups 퐻푖(퐶•) = ker(휕푖)/ im(휕푖+1). So, again, . The following definition first appeared in [15], we see two flavors of equivalence: one that produces iso- although we follow more closely the one given in [6]. after taking appropriate homotopy classes of ℳ maps, and one defined by an algebraic . Definition 8. A is a category together with a choice of three distinguished classes of morphisms: So what is the comparable kind of object to CW com- ∼ plexes here? The following theorem provides an answer. weak equivalences (→), cofibrations (↪), and fibrations (↠), each closed under composition and containing all isomor- Theorem 6. Any quasi-isomorphism between nonnegatively phisms and such that the following five axioms hold. graded chain complexes made up of projective modules is a chain (MC1) The category ℳ has all small limits and colimits. homotopy equivalence. (MC2) All three classes of morphisms are closed under But, as before, chain complexes of projective modules retracts. are also quite general. (MC3) The class of weak equivalences satisfies the two-out- of-three property: if 푓∶ 푋 → 푌 and 푔∶ 푌 → 푍 are Theorem 7. Any nonnegatively graded chain complex has a morphisms in ℳ such that two of 푓, 푔, and 푔푓 projective resolution, namely, a quasi-isomorphic chain complex are weak equivalences, then so is the third. made up entirely of projective modules. (MC4) Consider any commutative square Once again, we can define a category whose objects are / 퐴 _ > 푋 푅 } nonnegatively graded chain complexes of -modules and } whose maps are chain homotopy classes of maps between 푖 } 푝  }  projective resolutions. The resulting category is often / 퐵 푌. called the in . This second example suggests that the behavior of a ho- If either 푖 or 푝 is a weak equivalence, then a dotted motopy theory is not isolated to the world of topological arrow exists, making both triangular diagrams spaces but can occur in other . commute. (MC5) Any morphism 푓∶ 푋 → 푌 can be factored in two Model Categories ways: as ∼ In his work on [16], Quillen 푋 ↪ 푋′ ↠ 푌 observed further examples of a similar flavor, such as the category of topological spaces with rational equivalences and as ∼ and the category of differential graded Lie algebras with 푋 ↪ 푌′ ↠ 푌. appropriate quasi-isomorphisms. He proceeded to specify It is not immediately clear that these axioms provide axioms for a “homotopy theory” to capture this behavior; the desired structure that we specified above. Let us first the resulting structure is known as a model category. We give address the question of identifying suitably “nice” objects. a brief introduction to model categories here and refer the The fact that ℳ has limits and colimits implies, in particu- reader to [6] or [9] for more details. lar, that ℳ has both an initial object, which we denote by Before stating the definition, let us summarize the in- ∅, and a terminal object, which we denote by ∗. Recall gredients that seemed to make the examples from the pre- that an object ∅ of a category is initial if there is a unique vious work. First, we need a category together with morphism ∅ → 푋 for any object 푋 of ℳ. Dually, an ob- some morphisms that we have designated as “weak equiva- ject ∗ is terminal if there is a unique map 푋 → ∗ for any lences,” which should capture the desired notion of “same- 푋. ness” between objects. To construct the homotopy cate- gory, however, our examples suggest that we also need: Definition 9. An object 푋 of a model category ℳ is cofi- • some collection of objects that are sufficiently nice brant if the unique map ∅ → 푋 is a . It is fibrant and such that any object in the category is weakly if the unique map 푋 → ∗ is a . equivalent to one; We thus propose that the “nice” objects in a model cate- • a notion of homotopy of functions, at least be- gory are those that are both cofibrant and fibrant. To show tween “nice” objects; and that any object is weakly equivalent to such, we can apply • an agreement between “homotopy equivalence” (MC5) as follows. Given any object 푋, we can factor and “weak equivalence” on the “nice” objects. the map ∅ → 푋 as As Quillen realized, one can encode this information ∼ via the data of two additional classes of morphisms which, ∅ ↪ 푋푐 ↠ 푋,

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1425 and 푋푐 is a cofibrant replacement of 푋. Dually, 푋 has a There is also a model structure with the same weak fibrant replacement 푋푓, coming from the factorization equivalences in which the cofibrations are levelwise mono- morphisms and the fibrations are levelwise epimorphisms ∼ 푓 푋 ↪ 푋 ↠ ∗. that each has injective cokernel. In this structure, all ob- One can check that taking a fibrant replacement of a cofi- jects are cofibrant, and the fibrant objects are the chain 푐푓 complexes made up of injective modules. brant object, denoted by 푋 , produces an object that is still cofibrant, and similarly if we do the replacements in An important feature of model categories is that their the opposite order. homotopy categories do not depend on the model struc- A more sophisticated argument can be used to show ture used but only on the weak equivalences. Thus, if we that this structure allows for a well-defined “homotopy” have an underlying category and choice of weak equiva- between morphisms 푓∶ 푋 → 푌 in lences but two model structures with different fibrations ℳ for which 푋 and 푌 are both cofibrant and fibrant. We and cofibrations, then the resulting homotopy categories refer the reader to [6, §4] for a good treatment of this con- are suitably equivalent to one another. struction. However, different model categories, even with differ- Definition 10. Let ℳ be a model category. Define its ho- ent underlying categories, can still have equivalent motopy category Ho(ℳ) to have the same objects as ℳ and homotopy categories. We would like to specify such an to have morphisms between objects 푋 and 푌 defined by equivalence, and to do so we first consider what kind of between model categories preserves the relevant 푐푓 푐푓 HomHo(ℳ)(푋, 푌) = Homℳ(푋 , 푌 )/ ∼, structure. Since this structure appears in two dual flavors, namely, that of the cofibrations and that of the fibrations, where we identify maps in the same homotopy class. it is convenient to talk about an adjoint pair of In other words, maps 푋 → 푌 in the rather than a single functor between two model categories. are defined by the homotopy classes of maps between cofi- Recall that 퐹∶ ℳ → 풩 and 퐺∶ 풩 → ℳ are adjoint func- brant and fibrant replacements of 푋 and 푌. tors, with 퐹 the left adjoint and 퐺 the right adjoint, if there There is a natural functor ℳ → Ho(ℳ) that takes weak is an isomorphism equivalences of ℳ to isomorphisms in Ho(ℳ). Thus, the Hom풩(퐹푋, 푌) ≅ Homℳ(푋, 퐺푌) homotopy category is the context in which weak equiva- lences from ℳ become isomorphisms. that is natural for objects 푋 of ℳ and objects 푌 of 풩. Let Now that we have a definition of model categories, let us now incorporate model structures. us see how the examples we have considered fit into this framework. Definition 13. Let ℳ and 풩 be model categories. An adjoint pair of functors Example 11. The category of topological spaces with weak homotopy equivalences has a model structure. The fibra- 퐹∶ ℳ ⇄ 풩∶ 퐺 tions are the Serre fibrations, or continuous maps 푋 → 푌, is a Quillen pair if the left adjoint 퐹 preserves cofibrations such that, for any CW complex 퐴, a lift exists in any dia- and the right adjoint 퐺 preserves fibrations. gram of the form / To get an appropriate notion of equivalence between 퐴 × {0} t: 푋 t model categories, we incorporate weak equivalence data. t t  t  Definition 14. A Quillen pair 퐹∶ ℳ ⇄ 풩∶ 퐺 is a Quillen / 퐴 × [0, 1] 푌. equivalence when a morphism 퐹푋 → 푌, with 푋 cofibrant and 푌 fibrant, is a weak equivalence in 풩 precisely when The cofibrations are the retracts of inclusions given bycell its corresponding adjoint morphism 푋 → 퐺푌 is a weak attachments. With this structure, all objects are fibrant, equivalence in ℳ. and the cofibrant objects are the retracts of CW complexes. The following result establishes that this definition pro- Example 12. The category of nonnegatively graded chain duces the desired effect on the homotopy category. complexes with quasi-isomorphisms has a model structure in which the fibrations are levelwise epimorphisms and Proposition 15. If 퐹∶ ℳ ⇄ 풩∶ 퐺 is a Quillen equiva- the cofibrations are levelwise monomorphisms that each lence, then the induced adjoint pair has projective cokernel. Again, all objects are fibrant, and 퐹∶ Ho(ℳ) ⇄ Ho(풩)∶ 퐺 the cofibrant objects are precisely the chain complexes whose component modules are all projective. defines an equivalence of categories.

1426 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 Having a Quillen equivalence between model categories In an 푛-simplex, or its , observe that the 0- is stronger than simply having an equivalence between the simplices have a natural ordering, so we can label them by two homotopy categories, however. After giving an exam- 0, 1, … , 푛. We thus define the 푘-horn 푉[푛, 푘] by remov- ple, we will look more deeply into this structure. ing the face opposite the vertex labeled by 푘 from 휕Δ[푛] for any 0 ≤ 푘 ≤ 푛. When 푛 = 2, we can visualize 휕Δ[2] Simplicial Sets as Let us consider a classical example of two model categories @ 1 ?? that are Quillen equivalent to one another: the model ?? ?? structure on topological spaces that we have already seen ? / and a corresponding model structure on simplicial sets. 0 2. The latter are combinatorial objects of a similar flavor to, Then the horns 푉[2, 0], 푉[2, 1], and 푉[2, 2] can be de- but more general than, simplicial complexes. Aside from picted respectively as this relationship with topological spaces, simplicial sets 1 1 1 play a central role in the study of general homotopy the- ¨D ¨D 77 77 ¨¨ ¨¨ 77 77 ories. ¨¨ ¨¨ 77 77 ¨¨ ¨¨ 7 7 Definition 16. The category Δ has as objects the finite or- / / 0 2, 0 2, 0 2. dered sets [푛] = {0 ≤ 1 ≤ ⋯ ≤ 푛} and as morphisms the weakly order-preserving functions. Since simplicial sets are defined as functors, we can take Depicting only the generating morphisms for this cate- natural transformations between them to obtain a category 풮풮푒푡푠 gory, we can visualize Δ as of simplicial sets, which we denote by . More con- o / o / cretely, a map of simplicial sets 퐾 → 퐿 consists of maps [0] /[1]o /[2] ⋯ . 퐾푛 → 퐿푛 for all 푛 ≥ 0 that commute with the face and Given any category 풞, we can take its opposite category degeneracy maps appropriately. 풞op, which has the same objects as 풞 but in which the Geometric realization defines a functor direction of the morphisms is reversed. We thus consider | − |∶ 풮풮푒푡푠 → 풯표푝, the category Δop. where 풯표푝 denotes the category of topological spaces and op Definition 17. A is a functor 퐾∶ Δ → 풮푒푡푠. continuous maps. This functor has a right adjoint, the sin- We can thus think of a simplicial set as a diagram of sets gular functor o o / o / Sing∶ 풯표푝 → 풮풮푒푡푠, 퐾0 o 퐾1 o 퐾2 ⋯ 푛 defined by Sing(푋)푛 = Hom(Δ , 푋). satisfying certain relations. The maps 퐾푛 → 퐾푛−1 are called face maps and behave analogously to face maps be- We can use the geometric realization functor to define tween simplicial complexes. The main difference between a model structure on the category of simplicial sets. simplicial sets and simplicial complexes is that 푛-simplices, Definition 19. A map 퐾 → 퐿 of simplicial sets is a weak here the elements of the set 퐾푛, are not required to be gen- equivalence if its geometric realization |퐾| → |퐿| is a weak eralized triangles in terms of how their faces fit together. homotopy equivalence of topological spaces. The maps 퐾푛 → 퐾푛+1 are called degeneracy maps and pro- vide a specific way for an 푛-simplex to be thought of as a We define the cofibrations simply to be monomor- “degenerate” simplex in each higher . phisms of simplicial sets. To define the fibrations, we use Given a simplicial set 퐾, it can be geometrically realized the horn inclusions 푉[푛, 푘] → Δ[푛] from the above ex- to a topological space |퐾|. The idea is to take a geomet- ample. 푛 ric 푛-simplex Δ for each element of 퐾푛, for all 푛 ≥ 0, Definition 20. A map 퐾 → 퐿 of simplicial sets is a Kan and glue together simplices as specified by the face maps. fibration if, for any 푛 ≥ 1 and 0 ≤ 푘 ≤ 푛, a dotted arrow Thus, although they are defined combinatorially, we can lift exists in any commutative diagram of the form think of simplicial sets geometrically. Let us look at some // 푉[푛, 푘] w; 퐾 examples. w w 푛 ≥ 0 푛 w Example 18. For any , define the standard -simplex  w  Δ[푛] = Hom (−, [푛]) / to be the simplical set Δ . As its Δ[푛] 퐿. name suggests, its geometric realization is the usual topo- logical 푛-simplex Δ푛. If we remove the 푛-simplex given by Theorem 21. There is a model structure on the category of sim- the identity map in HomΔ([푛], [푛]), we obtain its bound- plicial sets with weak equivalences, fibrations, and cofibrations ary 휕Δ[푛]. defined as above.

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1427 Using these definitions, observe that every object is cofi- of categories with weak equivalences and no assumed addi- brant and that the fibrant objects are the Kan complexes or tional structure, sometimes referred to as relative categories. simplicial sets 퐾 for which the unique map 퐾 → Δ[0] is Definition 23. A category with weak equivalences is a cate- a Kan fibration. gory 풞 together with a subcategory 풲 whose morphisms The following theorem of Quillen [15], which is proved are called weak equivalences and satisfy the two-out-of-three concurrently with the one above, justifies our claim that property. simplicial sets are combinatorial models for topological spaces. Because the homotopy category does not really depend Theorem 22. The adjoint pair (|−|, Sing) defines a Quillen on a given model structure and because we do not need a equivalence between the model structures on simplicial sets and model structure to have the data of a category with weak topological spaces. equivalences anyway, we consider a category with weak equivalences to be the essential data of what we might call This result reinforces the idea that simplicial sets pro- a “homotopy theory.” vide a combinatorial model for topological spaces in that they have the same essential homotopy theory. A Homotopy Theory of Homotopy Theories Categories with Weak Equivalences and We are now in a position to investigate homotopy theories Localization as objects of study. Returning to the ideas from the be- ginning of this article, we could then ask, what are the ap- Unfortunately, not every category with a good notion of propriate functions between homotopy theories, and what weak equivalence actually has a model structure. Some- might a natural choice of equivalences be? We have seen times the problem is categorical, in that the underlying the answers to these questions in the setting of model cat- category does not have enough limits or colimits. In other egories, but what about for more general categories with situations it is difficult or even impossible to find appropri- weak equivalences? ate classes of fibrations and cofibrations that work well. So For functions (풞, 풲) → (풞′, 풲′), a natural choice one might ask, can we get by without a model structure? is to take functors 퐹∶ 풞 → 풟 such that 퐹(풲) ⊆ 풲′, Formally, the answer is yes. Given any category 풞 and so that weak equivalences are preserved. To answer the choice of class 풲 of morphisms in that category, we can question of what equivalences should be, let us first recall take a localization of 풞 by formally adjoining inverses to the definition of equivalence of categories. all morphisms of 풲, as well as all composites involving those new morphisms, to again obtain a category. Indeed, Definition 24. A functor 퐹∶ 풞 → 풟 is an equivalence of we can take the resulting category to be the homotopy cate- categories if: (풞, 풲) gory of the pair . This homotopy category is a local- (1) 퐹 is fully faithful, in that, for any objects 푥 and 푦 of 풞, (풞, 풲) 풞 ization of , in that any functor from to another the map category 풟 that takes weak equivalences of 풞 to isomor- phisms in 풟 must factor through the homotopy category. Hom풞(푥, 푦) → Hom풟(퐹푥, 퐹푦) Since that description seems so much more simple, why is a bijection of sets; and did we need to go to all the trouble of a model structure? (2) 퐹 is essentially surjective, in that, for any object 푧 of 풟, One reason is that the formal localization that we there is an object 푥 of 풞 such that 퐹푥 ≅ 푧. described here is often not set-theoretically well-behaved. Typically, we do not assume that a category has a set of The question of incorporating weak equivalence data objects, but can have a proper class, as we see in many into this definition turns out to be surprisingly delicate. common examples, such as the or the cate- For example, a natural generalization of essential surjectiv- gory of topological spaces. Consequently, the collection of ity would be that any object of 풟 is weakly equivalent to an morphisms often also forms a proper class. However, it is object in the image of 퐹. However, if “weakly equivalent” usually assumed, given two objects 푋 and 푌 of a category, is to be an equivalence relation, two objects can be weakly that Hom(푋, 푌) is a set rather than a class. The localiza- equivalent without having a single weak equivalence be- tion process that we have just described does not necessar- tween them, but instead have a zig-zag such as ily preserve this property; the output could be a structure ∼ ∼ ∼ • → • ← • → • with proper hom classes rather than hom sets. There are ways to work around this issue, for example, invoking pas- connecting them. sage to a higher set-theoretic universe. When we apply the This difficulty is not insurmountable, as shown in work homotopy category construction to a model category, we of Barwick and Kan [1], but here we instead look at an ear- avoid these issues and get a category in the preferred sense. lier approach to this difficulty implemented by Dwyer and Nonetheless, let us now turn to a more general theory Kan. Using the methods of simplicial localization, they

1428 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 repackage the data of a homotopy theory into that of a Theorem 27. Up to Dwyer–Kan equivalence, every simplicial simplicial category [5]. category can be obtained as the simplicial localization of a cat- egory with weak equivalences. Definition 25. A simplicial category is a category 풞 such that the morphisms between objects 푋 and 푌 form a sim- Thus, we can sensibly regard a simplicial category as a plicial set Map풞(푥, 푦) in a way compatible with composi- “homotopy theory” and the collection of all small simpli- tion and identity morphisms. A simplicial functor 풞 → 풟 cial categories, together with Dwyer–Kan equivalences, as preserves this simplicial structure. the “homotopy theory of homotopy theories.” In fact, we can say even more. For example, the category of simplicial sets forms a sim- plicial category, where Theorem 28. There is a model structure on the category of small simplicial categories in which the weak equivalences are Map(퐾, 퐿)푛 = Hom(퐾 × Δ[푛], 퐿). the Dwyer–Kan equivalences. The idea behind Dwyer and Kan’s simplicial localiza- We omit the details of the fibrations and cofibrations tion is that we can build a simplicial category 퐿(풞, 푊) here but refer the reader to [2]. from a category with weak equivalences (풞, 풲) by defin- Thinking of a homotopy theory as being modeled by a ing Map(푥, 푦) to have 0-simplices given by zig-zags such simplicial category, we can take the following alternative as ∼ perspective. Taking the components of mapping spaces in 푥 → • ← • → 푦 a simplicial category via the functor 휋0 results in an ordi- in which all left-going arrows are weak equivalences but nary category. We can thus think of any morphism in a which can be any length. Higher 푛-simplices are given by given mapping space as representing an hammocks of length 푛; an example when 푛 = 1 is in that category and any two such equivalent morphisms as being “homotopic” to one another. In this way, we can o ∼ ? • • @ think of a simplicial category as representing a “category  @@  @@ up to homotopy,” in the sense that morphisms can be de-  @@  @ formed in a manner analogous to of maps of 푥 ? ∼ ∼ 푦. topological spaces. ?? ~> ?? ~~ For many applications, it is useful to push this idea still ?? ~~ ?   ~~ further and ask that composition of morphisms be defined o ∼ • • only up to homotopy or that associativity of composition be required to hold only up to homotopy. It is this idea One can recover the homotopy category (or ordinary that we wish to explore in the next two sections. localization) by taking the sets of components of each of these mapping spaces. Here, additional structure of these What Is a Category Up to Homotopy? mapping spaces captures how much more information is Now that we have the concept of a category up to homo- present in a homotopy theory than in its associated homo- topy, let us develop it from a different angle, starting with topy category. the following way to think of an ordinary category as a sim- In the setting of simplicial categories, we have the fol- plicial set. lowing natural generalization of the notion of equivalence of categories. Definition 29. Let 풞 be a small category. Its is the simplicial set nerve(풞) given by Definition 26. A simplicial functor 푓∶ 풞 → 풟 is a Dwyer– Kan equivalence if: nerve(풞)푛 = Hom([푛], 풞). (1) for any pair of objects 푥, 푦 of 풞, the induced map of Observe that, in this definition, we are considering the simplicial sets ordered set [푛], which we previously regarded as an object of Δ, as a category with 푛 + 1 objects, depicted as Map풞(푥, 푦) → Map풟(푓푥, 푓푦) 0 → 1 → 2 → ⋯ → 푛. is a weak equivalence of simplicial sets; and Thus, the 0-simplices of the nerve correspond to the ob- (2) the functor 휋0풞 → 휋0풟 is essentially surjective, jects of 풞 and the 1-simplices to the morphisms. For 푛 ≥ where 휋0풞 denotes the category with the same objects 2, the set of 푛-simplices of the nerve of 풞 is precisely the as 풞 and Hom휋 풞(푥, 푦) = 휋0 Map (푥, 푦). 0 풞 set of chains of 푛 composable arrows in the category 풞. The following result of Dwyer and Kan from [4] tells A natural question, then, is whether the nerve functor us that simplicial categories always arise from simplicial takes equivalences of categories to weak equivalences of localization. categories, which is indeed the case.

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1429 Proposition 30. If 퐹∶ 풞 → 풟 is an equivalence of cate- other simplicial sets and in particular from nerves of more gories, then nerve(퐹)∶ nerve(풞) → nerve(풟) is a weak general categories. equivalence of simplicial sets. Let us return to our example of horns in a 2-simplex. What does being able to extend them in a 2-simplex tell However, the converse statement is false, as the follow- us? In the case of 푉[2, 1], a unique lifting corresponds ing example illustrates. to the existence of a composite 1-simplex, as we would ex- Example 31. Consider the category [0], which consists of pect in the nerve of a category. However, lifting along the one object and no nonidentity morphisms, and the cate- horns 푉[2, 0] and 푉[2, 2] does not give us composition gory [1], which consists of two objects and a single data but rather the existence of one-sided inverses. This nonidentity morphism between them. The natural inclu- example suggests the following result. [0] ↪ [1] sion is not an equivalence of categories, since Proposition 33. A simplicial set is the nerve of a category if [1] the object 1 of is not isomorphic to an object in the and only if it has unique liftings along the inner horn inclusions, image of this functor. However, its nerve is an inclusion namely, 푉[푛, 푘] → Δ[푛] for 0 < 푘 < 푛. Δ[0] → Δ[1] that after geometric realization is simply the inclusion of the endpoint of an interval. Since both We thus have good candidates for categories up to ho- of these spaces are contractible, this map is a weak equiva- motopy by removing the uniqueness assumption. lence. Definition 34. A quasi-category is a simplicial set 퐾 such The problem here arises because weak equivalences be- that a lift exists in any diagram of the form tween simplicial sets are defined in terms of geometric re- / 푉[푛, 푘] x; 퐾 alization, which forgets the directionality of 1-simplices in x x a simplicial set. More importantly for nerves of categories, x we lose information about whether a 1-simplex arises from  x an isomorphism or from a noninvertible morphism. This Δ[푛] data, however, is critical for verifying essential surjectivity. for every 푛 ≥ 2 and 0 < 푘 < 푛. We would like to have a more refined notion of weak equiv- alence that does not identify nerves of nonequivalent cat- We then describe our desired equivalences as the weak egories. equivalences in a model structure characterized by its nice We are not going to describe these new weak equiva- objects as follows [11]. lences explicitly, since it is quite technical to do so, but Theorem 35. There is a model structure on the category of rather use elements of the previously described model simplicial sets such that the fibrant and cofibrant objects are structure on simplicial sets to point toward them. In partic- precisely the quasi-categories. This model structure is Quillen ular, we look at the relationship between Kan complexes equivalent to the one for simplicial categories. and nerves of or categories with all morphisms invertible. This theorem serves to give us another model for the Recall from above that a Kan complex is a simplicial set homotopy theory of homotopy theories, but we also have 퐾 such that a dotted arrow lift exists in any diagram of the that the weak equivalences (often called Joyal equivalences) form between nerves of categories correspond exactly to equiva- / lences of categories. 푉[푛, 푘] x; 퐾 x x x AnotherApproachtoCategoriesUptoHomotopy  x Rezk developed a different strategy for defining categories Δ[푛] up to homotopy [17]. The idea is to keep the same notion where 푛 ≥ 1 and 0 ≤ 푘 ≤ 푛, making the diagram com- of weak equivalence between simplicial sets but to separate mute. out isomorphism data in the nerve of a category. We do this separation via an extra simplicial direction. Proposition 32. A simplicial set is the nerve of a if and only if it is a strict Kan complex, in that the lifts along the Definition 36. A bisimplicial set is a functor Δop → 풮풮푒푡푠. horn inclusions are unique. In other words, a bisimplicial set consists of a simplicial One can thus think of a Kan complex in which the lifts diagram of simplicial sets. To describe how to obtain one are not unique as a groupoid up to homotopy. Since every from a category, let us first give some notation. object in a model category is weakly equivalent to one that Let 풞 be a small category and let 푛 ≥ 0. Then the cate- is fibrant, we see our difficulty from another angle: weak gory of functors 풞[푛] has as objects the functors [푛] → 풞 equivalences cannot tell nerves of groupoids apart from and as morphisms the natural transformations between

1430 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 such functors. Given a small category 풟, denote by iso(풟) This result is proved in [10]; a comparison with simpli- the subcategory of 풟 consisting only of the isomorphisms cial categories can be found in [3]. of 풟. Other Perspectives 풞 Definition 37. Let be a small category. Its classifying There are also other ways to model such structures, includ- diagram 푁풞 is the bisimplicial set given by ing Segal categories [7], [14]; an appropriate notion of 퐴∞- [푛] (푁풞)푛 = nerve(iso(풞 )). categories [8]; the 1-complicial sets of Verity [13], [20]; and, going back to the origins of the subject, a model struc- Observe that (푁풞)0 is essentially the nerve of iso(풞), ture on the category of small categories with weak equiv- so contains only information about the isomorphisms of alences [1]. An axiomatic treatment for such models was 풞. Data about other morphisms on 풞 appears in (푁풞)1. given by Toën [19], and a very general approach using the We refer the reader to [17] for a more explicit version of framework of 2-categories is being developed by Riehl and the following result. Verity [18]. Proposition 38. A further point of view that we have not considered here, (1) For every 푛 ≥ 2, there is a natural isomorphism but which is nonetheless very important, is that homotopy theories, or categories up to homotopy, are a certain flavor (푁풞)푛 ≅ (푁풞)⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1 ×(푁풞)0 ⋯ ×(푁풞)0 (푁풞)1 . of higher categories called (∞, 1)-categories. An introduc- 푛 tory treatment to this point of view was presented in [12]. (2) The image of the degeneracy map (푁풞)0 → (푁풞)1 con- Whether regarded as a homotopy theory or as a higher cat- sists precisely of the information about isomorphisms in 풞. egory, these structures have become widely used, not only (3) The simplicial set (푁풞)∗,0 is precisely the nerve of 풞. in homotopy theory but in a range of subjects including It follows from this result that the classifying diagram mathematical physics and . does what we want: a functor 풞 → 풟 is an equivalence References 푛 ≥ 0 of categories if and only if, for every , the map [1] Barwick C, Kan DM. Relative categories: Another model (푁풞)푛 → (푁풟)푛 is a weak equivalence of simplicial sets. for the homotopy theory of homotopy theories, Indag. Thus, we have a way to describe categories in terms of bi- Math. (N.S.) 23 (2012), no. 1–2, 42–68. MR2877401 simplicial sets. To produce a model for categories up to [2] Bergner JE. A model category structure on the category of homotopy, we weaken the first two properties of the clas- simplicial categories, Trans. Amer. Math. Soc. 359 (2007), sifying diagram. 2043–2058. MR2276611 [3] Bergner JE. Three models for the homotopy theory Definition 39. of homotopy theories, 46 (2007), 397–436. (1) A bisimplicial set 푋 is a Segal space if, for each 푛 ≥ 2, MR2321038 the map [4] Dwyer WG, Kan DM. Equivalences between homotopy theories of diagrams, and algebraic 퐾- 푋 → 푋 ×ℎ ⋯ ×ℎ 푋 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟1 푋0 푋0 1 theory (Princeton, N.J., 1983), 180–205, Ann. of Math. 푛 Stud., 113, Princeton Univ. Press, Princeton, NJ, 1987. MR921478 is a weak equivalence of simplicial sets, where the [5] Dwyer WG, Kan DM. Function complexes in homotopi- right-hand side denotes an iterated homotopy pull- cal algebra, Topology 19 (1980), 427–440. MR584566 back. [6] Dwyer WG, Spalinski J. Homotopy theories and model (2) It is a complete Segal space if, additionally, the image of categories, Handbook of algebraic topology, Elsevier, 1995. the degeneracy map 푋0 → 푋1 is weakly equivalent to MR1361887 the subsimplicial set of homotopy equivalences. [7] Hirschowitz A, Simpson C. Descente pour les 푛-champs, preprint, available at math.AG/9807049, 1998. We again have an associated model structure that we [8] Horel G. The homotopy theory of 퐴∞-categories, describe in terms of the nice objects [17]. preprint, available at https://geoffroy.horel.org/ Theorem 40. There is a model structure on the category of Ainfty%20categories.pdf. bisimplicial sets in which all objects are cofibrant and the fibrant [9] Hovey M. Model categories, Mathematical Surveys and Monographs, 63, American Mathematical Society, 1999. objects are complete Segal spaces. MR1650134 Furthermore, property (3) from Proposition 38 above [10] Joyal A, Tierney M. Quasi-categories vs Segal spaces, Con- generalizes as follows. temp. Math. 431 (2007), 277–326. MR2342834 [11] Lurie J. Higher theory, Annals of Mathematics Stud- Theorem 41. If 푋 is a complete Segal space, then the simplicial ies, 170, Princeton University Press, Princeton, NJ, 2009. set 푋∗,0 is a quasi-category. Indeed, this assignment defines a MR2522659 Quillen equivalence between the two model structures.

OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1431 CALL FOR NOMINATIONS [12] Lurie J. What is … an ∞-category? Notices Amer. Math. Soc. 55 (2008), no. 8, 949–950. MR2441526 Nemmers Prize [13] Ozornova V, Rovelli M. Model structures for (∞, 푛)- categories on (pre)stratified simplicial sets and pre- in Mathematics stratified simplicial spaces, preprint, available at $200,000 math.AT/1809.10621, 2018. [14] Pelissier R. Cat´egoriesenrichies faibles, preprint, avail- able at math.AT/0308246, 2003. [15] Quillen D. Homotopical algebra, Lecture Notes in Math, 43, Springer-Verlag, Berlin-New York, 1967. MR0223432 Northwestern University invites nominations [16] Quillen D. Rational homotopy theory, Ann. of Math. (2) for the Frederic Esser Nemmers Prize in Math- 90 (1969), 205–295. MR0258031 ematics, to be awarded during the 2019–20 [17] Rezk C. A model for the homotopy theory of homo- academic year. The prize pays the recipient topy theory, Trans. Amer. Math. Soc. 353 (2001), 973–1007. $200,000. MR1804411 Candidacy for the Nemmers Prize is open to [18] Riehl E, Verity D. Elements of ∞-, www.math.jhu.edu/~eriehl/ those with careers of outstanding achievement preprint, available at elements.pdf. in their disciplines as demonstrated by major [19] Toën B. Vers une axiomatisation de la th´eorie des contributions to new knowledge or the develop- cat´egories sup´erieures, K-Theory 34 (2005), 233–263. ment of significant new modes of analysis. MR2182378 Individuals of all nationalities and institutional [20] Verity DRB. Weak complicial sets I: Basic homotopy the- affiliations are eligible except current or recent ory, Adv. Math. 219 (2008), 1081–1149. MR2450607 members of the Northwestern University faculty and recipients of the Nobel Prize. The 2020 Nemmers Prize recipient will deliver a public lecture and participate in other scholarly activities at Northwestern University for 10 weeks during the 2020–21 academic year. Nominations will be accepted until Decem- ber 1, 2019. The online submission form at nemmers.northwestern.edu requires the nomi­nee’s CV and one nominating letter of no more than 1,000 words describing the Julia E. Bergner no­minee’s professional experience, accom- Credits plishments, and qualifications for the award. Opener is courtesy of Getty Images. Nominations­ from experts in the field are Figure 1 was created by the AMS. preferred to institutional nominations; self- Author photo is courtesy of the author. nominations will not be accepted. Please email questions to [email protected].

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