equivalences in/of $(\infty,1)$-categories
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
According to the general pattern on (n,r)-category, an $(\infty,1)$-category is a (weak) ∞-category in which all $n$-morphisms for $n \geq 2$ are equivalences. This is the joint generalization of the notion of category and ∞-groupoid.
More precisely, this is the notion of category up to coherent homotopy: an $(\infty,1)$-category is equivalently
an internal category in ∞-groupoids/basic homotopy theory (as such usually modeled as a complete Segal space).
a category homotopy enriched over ∞Grpd (as such usually modeled as a Segal category).
Among all (n,r)-categories, $(\infty,1)$-categories are special in that they are the simplest structures that at the same time:
admit a higher version of category theory (limits, adjunctions, Grothendieck construction, etc, sheaf and topos theory, etc.) : (infinity,1)-category theory
and know everything about higher equivalences.
Notably for understanding the collections of all (n,r)-categories for arbitrary $n$ and $r$, which in general is an $(n+1,r+1)$-category, the knowledge of the underlying $(n,1)$- (and hence $(\infty,1)$-)category already captures much of the information of interest: it allows to decide if two given $(n,r)$-categories are equivalent and allows to obtain new $(n,r)$-categories from existing ones by universal constructions.
The collection of all $(\infty,1)$-categories forms the (∞,2)-category (∞,1)Cat.
There are a number of different ways to make the idea of an $(\infty,1)$-category precise, including quasi-categories, simplicially enriched categories, topologically enriched categories, Segal categories, complete Segal spaces, and $A_\infty$-categories (most of which can be done either simplicially or topologically). Additionally, any notion of ω-category can be specialized to a notion of $(\infty,1)$-category by simply requiring all $n$-cells for $n\gt 1$ to be invertible.
Unlike the case for general notions of $n$-category, almost all the definitions of $(\infty,1)$-category are known to form model categories that are Quillen equivalent. See also n-category for a summary of the state of the art about definitions of $n$-category and comparisons between them.
We start with the definition of “$(\infty,1)$-category“ that was promoted by Andre Joyal as a good model for the theory. This goes back to Boardman-Vogt in the 1970s and was further developed, by Jean-Marc Cordier and Tim Porter in the early 1980s.
This is a geometric definition of higher category which conceives an $(\infty,1)$-category as a simplicial set with extra property. It is a straightforward generalization of the definition of ∞-groupoid as a Kan complex, and, in fact, one alternative term used early on was ‘weak Kan complex’; see below.
Recall that a Kan complex is a simplicial set in which every horn $\Lambda^k[n]$, $0 \leq k \leq n$ has a filler. This condition may be read in words as: every collection of adjacent $n$-cells has a composite $n$-cell, even if the orientations of the cells don’t match. This implicitly encodes the invertibility of every cell: if the orientation does not match, we can invert the cell and then compose.
From this perspective one observes, by looking closely at the combinatorics, that the invertibility of the 1-cells in the simplicial set is enforced particularly by the condition that the outer horns $\Lambda^0[n]$ and $\Lambda^n[n]$ have fillers.
Therefore in a simplicial set in which only the inner horns $\Lambda^k[n]$ for $0 \lt k \lt n$ have fillers all cells are required to have a kind of inverse, except the 1-cells. (They may have inverses, too, but are not required to).
This is evidently a realization of the idea of an (n,r)-category with $n = \infty$ and $r = 1$.
Such a simplicial set with fillers for all inner horns
Boardman and Vogt called a weak Kan complex ;
Andre Joyal called a quasi-category;
Jacob Lurie called an $\infty$-category.
Here we follow Joyal and say quasi-category when we mean concretely the simplicial sets with extra property. We use the more general term “$(\infty,1)$-category“ for this or any of its equivalent models, discussed below, in order to distinguish from the term ∞-category or ω-category that is more traditionally understood to generically mean an $\infty$-category with no conditions on invertibility (in terms of (n,r)-category: an $(\infty,\infty)$-category).
With quasi-categories being just simplicial sets with extra property, there are evident and simple definitions of
the quasi-category of (∞,1)-functors| between two quasi-categories $C$ and $D$;
the quasi-category of all ∞-groupoids;
Similarly, Andre Joyal and Jacob Lurie have shown that all other constructions in category theory have good generalizations to quasi-categories, which usually have conceptually simple formulations: see Higher Topos Theory for more.
Despite the conceptual simplicity of quasi-categories, for computations and in particular for obtaining examples, it is often useful to pass to a slightly different model.
Recall that we said at the beginning that an $(\infty,1)$-category is supposed to be like an enriched category which is enriched over the category of ∞-groupoids. This turns out to make sense literally if one takes care to remember that $\infty$-groupoids themselves form a higher category.
As discussed at homotopy hypothesis there is a Quillen equivalence of the model categories of
the standard model structure on the nice category of compactly generated weakly Hausdorff topological spaces;
the standard model structure on the category of Kan complexes.
In fact, this is also equivalent to
If we take the notion of Kan complex to be the most manifest incarnation of the idea “∞-groupoid”, then under these equivalences one may think of
a simplicial set as representing the Kan complex which is obtained from it by “freely throwing in the missing inverses” of cells (technically: as representing its fibrant replacement);
a topological space $X$ as representing the Kan complex $\Pi(X)$, whose
0-cells are the points of $X$;
1-cells are the paths in $X$;
2-cells are the triangles in $X$;
etc.
With this interpretation understood (i.e. with these model structures understood), SSet-enriched categories do model $(\infty,1)$-categories.
For more see
A homotopical category is a category $C$ equipped with a class $W$ of weak equivalences. Every homotopical category $(C,W)$ has a quasi-localisation $C[W(-1)]$ which is a quasi-category. The simplicial set $C[W(-1)]$ is obtained from the nerve of $C$ by freely gluing a homotopy inverse to each morphism in $W$, and then, by adding simplices to turn it into a quasi-category (this last step is called a fibrant completion).
The quasi-category $C[W(-1)]$ is equivalent to the Dwyer-Kan localisation of $C$ with respect to $W$, via the equivalence between quasi-categories and simplicial categories mentioned above.
Conversely, every quasi-category is equivalent to the quasi-localisation of a homotopical category. This gives a representation of all $(\infty,1)$-categories in terms of homotopical categories. It follows that many aspects of the theory of $(\infty,1)$-categories can be expressed in terms of category theory.
When the homotopical category (C,W) is obtained from a Quillen model structure (by forgetting the cofibrations and the fibrations) the quasi-category C[W^(-1)] has finite limits and colimits. Conversely, I conjecture that every quasi-category with finite limits and colimits is equivalent to the quasi-localisation of a model category. In fact, every locally presentable quasi-category is a quasi-localisation of a combinatorial model by a result of Lurie. More can be said: the underlying category can taken to be a category of presheaves by a result of Daniel Dugger.
http://arxiv.org/abs/math/0007070
A specific notion of homotopical category is that of a model category. $(\infty,1)$-categories obtained as the Dwyer-Kan simplicial localizations of model categories have for instance finite $(\infty,1)$-limits and $(\infty,1)$-colimits. The locally presentable (∞,1)-categories are precisely those presented this way by combinatorial model categeories.
At the very beginning, a model category was understood as a “model for the category Top of topological spaces,” or more precisely homotopy types: some category with extra structure and properties which allows one to perform all operations familiar of the homotopy theory of topological spaces.
As mentioned above, from the point of view of (∞,1)-categories, Top may naturally be regarded an as (∞,1)-category and is in fact the archetypical example, analogous to how Set is the archetypical example of an ordinary category.
This indicates that, more generally, a model category should actually be a means to model (i.e. encode) in 1-categorical terms an $(\infty,1)$-category, and of course this is true since indeed any category with weak equivalences presents an $(\infty,1)$-category via Dwyer-Kan simplicial localization. In the case of a model category, however, or at least a simplicial model category, this $(\infty,1)$-category has a different, simpler construction.
A simplicial model category $\mathbf{A}$ is, in particular, a simplicially enriched category.
the full SSet-subcategory $\mathbf{A}^\circ$ on the fibrant-cofibrant objects of $\mathbf{A}$ happens to be Kan complex-enriched;
the homotopy coherent nerve $N(\mathbf{A}^\circ)$ of $\mathbf{A}^\circ$ is the quasi-category presented by $A$.
Up to equivalence, this gives the same $(\infty,1)$-category as the Dwyer-Kan hammock localization. With the relation between simplicially enriched categories and quasi-categories via homotopy coherent nerve understood, we shall here often not distinguish between $\mathbf{A}^\circ$ and $N(\mathbf{A}^\circ)$ as the $(\infty,1)$-category presented by a model category $A$.
Other models for $(\infty,1)$-categories are
Segal categories can be thought of as categories which are weakly enriched in topological spaces/simplicial sets/Kan complexes, where the definition of “weak” makes use of the notion of homotopy and homotopy limit in Top or SSet.
Complete Segal spaces are like internal categories in an (∞,1)-category.
This construction principle in particular lends itself to iteration and hence to an inductive definition of (∞,n)-category via Segal n-categories and n-fold complete Segal spaces.
An $A_\infty$-category can also be thought of as a category “weakly enriched” in spaces (i.e. $\infty$-groupoids), except that in contrast to the Segal approaches the “weakness” is specified algebraically and parametrized by an operad. This approach can be generalized to the Trimble definition of $n$-category or $(\infty,n)$-category.
A crucial point about the notion of $(\infty,1)$-category is that it supports all the standard constructions and theorems of category theory, if only the consistent replacements are made (isomorphism becomes equivalence, etc.).
The collection of all $(\infty,1)$-categories forms an (∞,2)-category called (∞,1)Cat.
Often it is useful to regard that as a (large) $(\infty,1)$-category itself, by discarding the non-invertible natural transformations.
There is a wealth of different presentations of $(\infty,1)$-categories.
See table - models for (∞,1)-categories.
In practice, it can be useful to be able to treat all “presentations of $(\infty,1)$-categories” on the same equal footing (e.g. relative categories and topologically-enriched categories). While truly model-independent foundations of $(\infty,1)$-category theory do not (yet) exist, this can be accomplished within any model of $(\infty,1)$-categories, which we proceed to describe. As quasicategories are by far the most well-developed, we use them as an ambient framework. We also take care to make as few choices (even “contractible” ones) as possible. However, we do not explicitly mention set-theoretic issues, though these are easily handled using Grothendieck universes.
Consider the $Kan$-enriched category $\underline{QCat}$ of quasicategories; for quasicategories $C$ and $D$, the Kan complex of morphisms between them is $\underline{hom}_{\underline{QCat}} = \iota(\underline{hom}_{sSet}(C,D))$, the largest Kan complex contained in their internal hom simplicial set.
Define a relative quasicategory to be a quasicategory equipped with a full sub-quasicategory of “weak equivalences” containing all equivalences. For relative quasicategories $(C,W_C)$ and $(D,W_D)$, write $\underline{hom}_{\underline{RelQCat}}((C,W_C),(D,W_D)) \subset \underline{hom}_{\underline{QCat}}(C,D)$ for the sub-Kan complex consisting of those maps which take $W_C$ into $W_D$. Note that using this definition, this is actually the inclusion of a disjoint union of connected components among Kan complexes (in the strictest possible sense).
There is an evident inclusion $min : \underline{QCat} \to \underline{RelQCat}$, which takes a quasicategory $C$ to the relative quasicategory $(C,C^\simeq)$.
Although a Quillen equivalence $M_1 \rightleftarrows M_2$ between model categories determines an equivalence of homotopy categories, note that neither adjoint functor need preserve weak equivalences. On the other hand, the restrictions $M_1^c \hookrightarrow M_1 \rightarrow M_2$ and $M_1 \leftarrow M_2 \hookleftarrow M_2^f$ (to the cofibrant objects of $M_1$ and the fibrant objects of $M_2$) do preserve weak equivalences, and these determine a hexagonal diagram of weak equivalences between relative categories (in the Barwick–Kan model structure), as in MazelGee16, Figure 1.
Using the previous observation, expand the diagram in the introduction of BarwickSchommerPries (relating a great many Quillen equivalent model categories presenting “the homotopy theory of $(\infty,1)$-categories”) into a diagram of weak equivalences between relative categories. As relative categories are particular examples of relative quasicategories, this defines a functor $F : K \to \underline{RelQCat}$ among fibrant objects of $(Cat_{sSet})_{Bergner}$.
Now, apply the right Quillen equivalence $N^{hc} : (Cat_{sSet})_{Bergner} \to sSet_{Joyal}$ (the homotopy-coherent nerve) to this cospan $\underline{QCat} \xrightarrow{min} \underline{RelQCat} \leftarrow K$.
The morphism $N^{hc}(min)$ of quasicategories admits a contractible Kan complex worth of quasicategorical left adjoints, any of which presents the localization of relative quasicategories. Choose one, and denote this quasicategorical adjunction by $L : N^{hc} (\underline{RelQCat}) \rightleftarrows N^{hc} ( \underline{QCat}) : min$.
It follows from the main theorem of Toen that the composite map $L \circ N^{hc}(F) : N(K) \cong N^{hc}(K) \to N^{hc}(\underline{RelQCat}) \to N^{hc}(\underline{QCat})$ is “essentially contractible” in the quasicategorical sense. More precisely, for any cofibration into an acyclic object $i : N(K) \to K' \approx pt$ in $sSet_{Joyal}$, there exists a contractible Kan complex worth of extensions of $L \circ N^{hc}(F)$ over $i$.
Define $(The(\infty,1)Cats) \subset N^{hc}(\underline{QCat})$ to be the maximal sub-Kan complex generated by the image of $L \circ N^{hc}(F)$. We write $Cat_{(\infty,1)} \in (The(\infty,1)Cats)$ for any vertex, and propose that to work “model-independently” is to work within $Cat_{(\infty,1)}$.
This sequence of maneuvers balances twin aims. On the one hand, Toen’s theorem asserts that after choosing a basepoint, this Kan complex is a model for $B(\mathbb{Z}/2)$. Thus, any sort of object which might be considered as “a presentation of an $(\infty,1)$-category” canonically determines an object of $Cat_{(\infty,1)}$ (where “canonical” must still be taken in the quasicategorical sense). On the other hand, it is completely independent of which vertex of $(The(\infty,1)Cats)$ we choose.
This diagram, taking place in $N^{hc}(\underline{QCat})$, elaborates on certain salient aspects of the passage from models of $(\infty,1)$-categories to a model-independent approach. (For a small amount of explanation of this diagram, see here.)
(∞,1)-category, internal (∞,1)-category, ∞-groupoid
For several years Andre Joyal – who was one of the first to promote the idea that for studying higher category theory it is good to first study $(\infty,1)$-categories in terms of quasi-categories – has been preparing a textbook on the subject. This still doesn’t quite exist, but an extensive write-up of lecture notes does:
Further notes (where the term “logos” is used instead of quasi-category) are in
Meanwhile Jacob Lurie, building on Joyal’s work, has considerably pushed the theory further. A comprehensive discussion of the theory of $(\infty,1)$-categories in terms of the models quasi-category and simplicially enriched category is
An brief exposition from the point of view of algebraic topology is in
A useful comparison of the four model category structures on
is in
More discussion of the other two models can be found at
and in the references listed at (∞,n)-category.
The relation between quasi-categories and simplicially enriched categories was discussed in detail in
Dan Dugger, David Spivak, Rigidification of quasi-categories (arXiv:0910.0814)
Dan Dugger, David Spivak, Mapping spaces in quasi-categories (arXiv:0911.0469)
The presentation of $(\infty,1)$-categories by homotopical categories and model categories is discussed in
A model by stratified spaces is in
A more model-independent abstract formulation is discussed in
An introduction to higher category theory through $(\infty,1)$-categories is
A foundational set of lecture notes is developing in
A survey with an eye towards higher algebra is in
Lecture notes are in
Dylan Wilson, Lectures on higher categories (pdf)
See also