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(infinity,1)-category

Idea

According to the general pattern on (n,r)-category, an $(\mathrm{\infty }\mathrm{,1})$-category is a (weak) ∞-category in which all $n$-morphisms for $n\ge 2$ are equivalences.

To some extent an $(\mathrm{\infty }\mathrm{,1})$-category can be thought of as a category enriched in (∞,0)-categories, namely in ∞-groupoids.

Among all (n,r)-categories, $(\mathrm{\infty }\mathrm{,1})$-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 $(n\mathrm{,1})$- (and hence $(\mathrm{\infty }\mathrm{,1})$-)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 $(\mathrm{\infty }\mathrm{,1})$-categories forms the (∞,2)-category (∞,1)Cat.

Definitions

There are a number of different ways to make the idea of an $(\mathrm{\infty }\mathrm{,1})$-category precise, including quasi-categories, simplicially enriched categories, topologically enriched categories, Segal categories, complete Segal spaces, and $A_{\mathrm{\infty }}$-categories (most of which can be done either simplicially or topologically). Additionally, any notion of ω-category can be specialized to a notion of $(\mathrm{\infty }\mathrm{,1})$-category by simply requiring all $n$-cells for $n>1$ to be invertible.

Unlike the case for general notions of $n$-category, almost all the definitions of $(\mathrm{\infty }\mathrm{,1})$-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.

Quasi-categories

We start with the definition of “$(\mathrm{\infty }\mathrm{,1})$-category” that was promoted by Andre Joyal as a good model for the theory. This goes back to Boardman-Vogt.

This is a geometric definition of higher category which conceives an $(\mathrm{\infty }\mathrm{,1})$-category as a simplicial set with extra property. It is a straightforward generalization of the definition of ∞-groupoid as a Kan complex.

Recall that a Kan complex is a simplicial set in which every horn ${\Lambda }^k[n]$, $0\le k\le n$ 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 horn|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 $0<k<n$ 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 $n=\mathrm{\infty }$ 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 $\mathrm{\infty }$-category.

Here we follow Joyal and say quasi-category when we mean concretely the simplicial sets with extra property. We use more generally term “$(\mathrm{\infty }\mathrm{,1})$-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 $\mathrm{\infty }$-category with no conditions on invertibility (in terms of (n,r)-category: an $(\mathrm{\infty },\mathrm{\infty })$-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;

• the quasi-category of all quasi-categories.

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.

$\mathrm{Top}$-, $\mathrm{Kan}$- and simplicially enriched categories

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 $(\mathrm{\infty }\mathrm{,1})$-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 $\mathrm{\infty }$-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

• the standard model structure on the category SSet of simplicial sets.

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 $(\mathrm{\infty }\mathrm{,1})$-categories.

For more see

• relation between quasi-categories and simplicial categories

Homotopical categories

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 simplicies 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 simplicial categories mentioned above.

Conversely, every quasi-category is equivalent to the quasi-localisation of a homotopical category. This gives a representation of all $(\mathrm{\infty }\mathrm{,1})$-categories in terms of homotopical categories. It follows that many aspects of the theory of $(\mathrm{\infty }\mathrm{,1})$-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 CW^(-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

Model categories

A specific notion of homotopical category is that of a model category. $(\mathrm{\infty }\mathrm{,1})$-categories obtained as the Dwyer-Kan simplicial localizations of model categories have for instance finite $(\mathrm{\infty }\mathrm{,1})$-limits and $(\mathrm{\infty }\mathrm{,1})$-colimit. 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 examples 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 $(\mathrm{\infty }\mathrm{,1})$-category. This indeed turns out to be true: there is a precise sense in which every model category presents an $(\mathrm{\infty }\mathrm{,1})$-category.

• given a model category $A$;

• it becomes canonically an SSet-enriched category making it a simplicial model category $A$;

• the full SSet-subcategory $A^{\circ }$ on the fibrant-cofibrant objects of $A$ happens to be Kan complex-enriched;

• the homotopy coherent nerve $N(A^{\circ })$ of $A^{\circ }$ is the quasi-category presented by $A$.

With the relation between simplical categories and quasi-categories via homotopy coherent nerve understood, we shall here often not distinguish between $A^{\circ }$ and $N(A^{\circ })$ as the $(\mathrm{\infty }\mathrm{,1})$-category presented by the model category $A$.

Segal categories and complete Segal spaces

Other models for $(\mathrm{\infty }\mathrm{,1})$-categories are

• Segal categories;

• complete Segal spaces.

These notions 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.

(Actually, complete Segal spaces are actualy more like internal categories in spaces which satisfy an extra “completeness” condition specifying that the extra data are redundant. But including this extra data turns out to be technically convenient in many ways.)

This construction principle in particular lends itself to iteration and hence to an inductive definition of (∞,n)-category.

$A_{\mathrm{\infty }}$-categories

An $A_{\mathrm{\infty }}$-category can also be thought of as a category “weakly enriched” in spaces (i.e. $\mathrm{\infty }$-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 $(\mathrm{\infty },n)$-category.

References

For several years Andre Joyal – who was the one to promote the idea that for studying higher category theory it is good to first study $(\mathrm{\infty }\mathrm{,1})$-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:

• Andre Joyal, The theory of quasicategories and its applications lectures at Simplicial Methods in Higher Categories, (pdf)

Meanwhile Jacob Lurie, building on Joyal’s work, has considerably pushed the theory further. A comprehensive discussion of the theory of $(\mathrm{\infty }\mathrm{,1})$-categories in terms of the models quasi-category and simplicially enriched category is

• Jacob Lurie, Higher Topos Theory .

A useful comparison of the four model category structures on

• quasi-categories;

• simplicially enriched categories;

• Segal categories;

• complete Segal spaces.

is in

• Julie Bergner, A survey of $(\mathrm{\infty }\mathrm{,1})$-categories (arXiv)

More discussion of the other two models can be found at

• Jacob Lurie, On the Classification of Topological Field Theories

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)