structures in a cohesive (∞,1)-topos
According to the general pattern on (n,r)-category, an -category is a (weak) ∞-category in which all -morphisms for are equivalences. This is the joint generalization of the notion of category and ∞-groupoid.
Among all (n,r)-categories, -categories are special in that they are the simplest structures that at the same time:
and know everything about higher equivalences.
Notably for understanding the collections of all (n,r)-categories for arbitrary and , which in general is an -category, the knowledge of the underlying - (and hence -)category already captures much of the information of interest: it allows to decide if two given -categories are equivalent and allows to obtain new -categories from existing ones by universal constructions.
There are a number of different ways to make the idea of an -category precise, including quasi-categories, simplicially enriched categories, topologically enriched categories, Segal categories, complete Segal spaces, and -categories (most of which can be done either simplicially or topologically). Additionally, any notion of ω-category can be specialized to a notion of -category by simply requiring all -cells for to be invertible.
Unlike the case for general notions of -category, almost all the definitions of -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 -category and comparisons between them.
We start with the definition of “-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 -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 , has a filler. This condition may be read in words as: every collection of adjacent -cells has a composite -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 and have fillers.
Therefore in a simplicial set in which only the inner horns for 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 and .
Such a simplicial set with fillers for all inner horns
Boardman and Vogt called a weak Kan complex ;
Jacob Lurie called an -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 “-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 -category with no conditions on invertibility (in terms of (n,r)-category: an -category).
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 -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 -groupoids themselves form a higher category.
In fact, this is also equivalent to
0-cells are the points of ;
1-cells are the paths in ;
2-cells are the triangles in ;
With this interpretation understood (i.e. with these model structures understood), SSet-enriched categories do model -categories.
For more see
A homotopical category is a category equipped with a class of weak equivalences. Every homotopical category has a quasi-localisation which is a quasi-category. The simplicial set is obtained from the nerve of by freely gluing a homotopy inverse to each morphism in , and then, by adding simplices to turn it into a quasi-category (this last step is called a fibrant completion).
Conversely, every quasi-category is equivalent to the quasi-localisation of a homotopical category. This gives a representation of all -categories in terms of homotopical categories. It follows that many aspects of the theory of -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.
A specific notion of homotopical category is that of a model category. -categories obtained as the Dwyer-Kan simplicial localizations of model categories have for instance finite -limits and -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 -category, and of course this is true since indeed any category with weak equivalences presents an -category via Dwyer-Kan simplicial localization. In the case of a model category, however, or at least a simplicial model category, this -category has a different, simpler construction.
Up to equivalence, this gives the same -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 and as the -category presented by a model category .
Other models for -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.
An -category can also be thought of as a category “weakly enriched” in spaces (i.e. -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 -category or -category.
A crucial point about the notion of -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.).
Often it is useful to regard that as a (large) -category itself, by discarding the non-invertible natural transformations.
There is a wealth of different presentations of -categories.
In practice, it can be useful to be able to treat all “presentations of -categories” on the same equal footing (e.g. relative categories and topologically-enriched categories). While truly model-independent foundations of -category theory do not (yet) exist, this can be accomplished within any model of -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 -enriched category of quasicategories; for quasicategories and , the Kan complex of morphisms between them is , 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 and , write for the sub-Kan complex consisting of those maps which take into . 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 , which takes a quasicategory to the relative quasicategory .
Although a Quillen equivalence between model categories determines an equivalence of homotopy categories, note that neither adjoint functor need preserve weak equivalences. On the other hand, the restrictions and (to the cofibrant objects of and the fibrant objects of ) 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 -categories”) into a diagram of weak equivalences between relative categories. As relative categories are particular examples of relative quasicategories, this defines a functor among fibrant objects of .
Now, apply the right Quillen equivalence (the homotopy-coherent nerve) to this cospan .
The morphism 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 .
It follows from the main theorem of Toen that the composite map is “essentially contractible” in the quasicategorical sense. More precisely, for any cofibration into an acyclic object in , there exists a contractible Kan complex worth of extensions of over .
Define to be the maximal sub-Kan complex generated by the image of . We write for any vertex, and propose that to work “model-independently” is to work within .
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 . Thus, any sort of object which might be considered as “a presentation of an -category” canonically determines an object of (where “canonical” must still be taken in the quasicategorical sense). On the other hand, it is completely independent of which vertex of we choose.
This diagram, taking place in , elaborates on certain salient aspects of the passage from models of -categories to a model-independent approach. (For a small amount of explanation of this diagram, see here.)
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 -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:
Meanwhile Jacob Lurie, building on Joyal’s work, has considerably pushed the theory further. A comprehensive discussion of the theory of -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
More discussion of the other two models can be found at
and in the references listed at (∞,n)-category.
A model by stratified spaces is in
A more model-independent abstract formulation is discussed in
An introduction to higher category theory through -categories is
A survey with an eye towards higher algebra is in
Lecture notes are in