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An ∞-Category? Jacob Lurie

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An ∞-Category? Jacob Lurie WHAT IS... ? an ∞-Category? Jacob Lurie ∗ One of the most celebrated invariants in alge- map f : Singn X → Singm X, given by composition braic topology is the fundamental group : given with a linear map of simplices ∆m → ∆n. For exam- a topological space X with a base point x, the ple, if we take f : {0, 1,...,n − 1}→{0, 1,...,n} fundamental group π1(X, x) is defined to be the to be the injective map that omits the value i, set of paths in X from x to itself, taken modulo then we obtain a map di : Singn X → Singn−1 X, homotopy. If we do not want to choose abasepoint which carries an n-simplex of X to its ith face. x ∈ X, we can consider instead the fundamental The simplicial set Sing X is sometimes called the groupoid π≤1X: this is a category whose objects singular complex of the topological space X. are the points of X, in which morphisms are given It turns out that the answer to question (B) is by paths between points (again taken modulo ho- “essentially everything”, at least to an algebraic motopy). The fundamental groupoid of X contains topologist. More precisely, we can use the singular slightly more information than the fundamental complex Sing X to construct a topological space group of X: it determines the set of path com- that is (weakly) homotopy equivalent to the origi- ponents π X (these are the isomorphism classes 0 nal space X. Consequently, no information is lost of objects in π X) and also the fundamental ≤1 by discarding the original space X and working group of each component (the fundamental group directly with the simplicial set Sing X. In fact, it is π (X, x) is the automorphism group of x in the 1 possible to develop the theory of algebraic topolo- category π X). The language of category theory ≤1 gy in entirely combinatorial terms, using simplicial allows us to package this information together in sets as surrogates for topological spaces. However, a very convenient form. not every simplicial set S behaves like the singular The fundamental groupoid π≤1X does not con- tain any information about the higher homotopy complex of a space; it is therefore necessary to single out a class of “good” simplicial sets to groups {πn(X, x)}n≥2. In order to recover infor- mation of this sort, it is useful to consider not work with. For this, we need to introduce a bit of only points and paths in X, but also the set of terminology. ∆n Let S = {Sn}n≥0 be a simplicial set. Here we continuous maps Singn X = Hom( , X) for ev- ery nonnegative integer n; here ∆n denotes the imagine that Sn denotes the set of all continuous n topological n-simplex. This raises three questions: maps from an n-simplex ∆ into a topological (A) What kind of a mathematical object is the space X, although such a space need not exist. For 0 ≤ i ≤ n, we define another set Λn(S), which we collection of sets {Sing X}n≥0? i n ∆n (B) How much does this object know about X? think of as a set of partially defined maps from (C) To what extent does this object behave into X: namely, maps whose domain consists of all n like a category? faces of ∆ that contain the ith vertex (this subset ∆n To address question (A), we note that Sing X = of is sometimes called an i-horn). Formally, we Λn define i (S) to be the collection of all sequences {Singn X}n≥0 has the structure of a simplicial set. That is, for every nondecreasing function f : {σj }0≤j≤n,j≠i of elements of Sn−1, which “fit togeth- {0, 1,...,m}→{0, 1,...,n}, there is an induced er” in the following sense: if 0 ≤ j < k ≤ n and j ≠ i ≠ k, then dj σk = dk−1σj ∈ Sn−2. There is a Λn Jacob Lurie is professor of mathematics at the Mas- restriction map Sn → i (S), given by the formula sachusetts Institute of Technology. His email address is τ ֏ (d0τ, d1τ,...,di−1τ, di+1τ,...dnτ). [email protected]. September 2008 Notices of the AMS 949 Definition 1. A simplicial set S ={Sn}n≥0 is a Kan category. The objects of S are the elements of S0, Λn complex if the map Sn → i (S) is surjective for all and the morphisms of S are the elements of S1. 0 ≤ i ≤ n. Suppose we are given two morphisms f , g ∈ S1 that are “composable” (that is, the source of g co- In other words, S is a Kan complex if every horn incides with the target of f ), as depicted in the fol- σ ∈ Λn(S) can be “filled” to an n-simplex of S. i lowing diagram Roughly speaking, a Kan complex is a simplicial set that resembles the singular complex of a topo- y logical space. In particular, the singular complex f g Sing X of a topological space X is always a Kan h complex. x z. We now address question (C): in what sense does the singular complex Sing X behave like a The morphisms f and g determine a horn σ0 ∈ category? To answer this, we observe that there is Λ2 1(S). If S is an ∞-category, then this horn can be a close connection between categories and simpli- filled to obtain a 2-simplex σ ∈ S2. We can then cial sets. For every category C and every integer define a new morphism h : x → z by passing to a n ≥ 0, let Cn denote the set of all composable face of σ ; this morphism can be viewed as a com- chains of morphisms position of g and f . The 2-simplex σ is generally = ◦ C0 → C1 → ... → Cn not unique, so that the composition h g f is not unambigiously defined. However, one can use the of length n. The collection of sets {C } has the n n≥0 horn-filling conditions of Definition 3 (for n > 2) structure of a simplicial set, which is called the to show that h is well-defined “up to homotopy”. nerve of the category C, and it determines C up This turns out to be good enough: there is a robust to isomorphism. For example, the objects of C are theory of ∞-categories, which contains generaliza- simply the elements of C , and the morphisms in 0 tions of most of the basic ideas of category theory C are the elements of C . 1 (limits and colimits, adjoint functors, ...). We may therefore view the notion of a simplicial Definition 3 is really only a first step towards a set as a generalization of the notion of a category. much more ambitious generalization of classical How drastic is this generalization? In other words, category theory: the theory of higher categories. how can we tell if a simplicial set arises as the One can think of ∞-categories as higher categories nerve of a category? The following result provides in which every k-morphism is required to be in- an answer: vertible for k > 1. = { } Proposition 2. Let S Sn n≥0 be a simplicial set. Example 5. For every topological space X, the C Then S is isomorphic to the nerve of a category if singular complex Sing X is an ∞-category. Since → Λn and only if, for each 0 <i<n, the map Sn i (S) Sing X determines the space X up to (weak) bijective. homotopy equivalence, we can view the theory The hypothesis of Proposition 2 resembles the of ∞-categories as a generalization of classical definition of a Kan complex but is different in homotopy theory. two important respects. Definition 1 requires that Examples 4 and 5 together convey the spirit ∈ Λn every horn σ i (S) can be filled to an n-simplex of the subject: the theory of ∞-categories can of S. Proposition 2 requires this condition only in be viewed as a combination of category theo- the case 0 <i<n but demands that the n-simplex ry and homotopy theory and has the flavor of be unique. Neither condition implies the other, but both. Where classical category theory provides they admit a common generalization: a language for the study of algebraic structures Definition 3. A simplicial set S is an ∞-category (groups, rings, vector spaces, ...), the theory of Λn ∞-categories provides an analogous language for if, for each 0 <i<n, the map Sn → i (S) is surjective. describing their homotopy-theoretic counterparts (loop spaces, ring spectra, chain complexes, ...). The notion of an ∞-category was originally The power of this language is only beginning to introduced (under the name weak Kan complex) be exploited. by Boardman and Vogt, in their work on homo- topy invariant algebraic structures (see [1]). The Further Reading theory has subsequently been developed more ex- [1] J. M. Boardman and R. M. Vogt, Homotopy In- tensively by Joyal (who refers to ∞-categories as variant Structures on Topological Spaces, Lecture quasicategories); a good reference is [2]. Notes in Mathematics 347, Springer-Verlag, Berlin and New York, 1973. Example 4. Let C be a category. Then the nerve [2] A. Joyal, Quasi-categories and Kan complexes, {C } ∞ C n n≥0 is an -category, which determines up Journal of Pure and Applied Algebra 175 (2002), to isomorphism. Consequently, we can think of an 207-222. ∞-category S = {Sn}n≥0 as a kind of generalized 950 Notices of the AMS Volume 55, Number 8.
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