nLab
infinity-category

Idea

$\mathrm{\infty }$-Categories are the entities studied in higher category theory. See also there.

In generalization to how an ordinary category has morphisms going between objects, and a 2-category has both morphisms (or 1-morphisms) between objects and 2-morphisms (or 2-cells) going between 1-morphisms, in an $\mathrm{\infty }$-category (sometimes called an $\omega$-category), there are $j$-morphisms going between $(j-1)$-morphisms for all $j=1,2,\dots$. (The $0$-morphisms are the objects of the $\mathrm{\infty }$-category.)

If all the $j$-morphisms in an $\mathrm{\infty }$-category are equivalences in some suitable sense, we call the $\mathrm{\infty }$-category an $\mathrm{\infty }$-groupoid. In this case we can think of the $j$-morphisms for $j\ge 1$ as “homotopies” and the $\mathrm{\infty }$-groupoid as a model for a “space.” By analogy, we can, if we wish, think of an arbitrary $\mathrm{\infty }$-category as a combinatorial model for a directed space containing higher directed homotopies.

There are many different definitions realizing the general idea of $\mathrm{\infty }$-category. Models for $\mathrm{\infty }$-categories usually fall into two classes:

• in the geometric definition of higher category an $\mathrm{\infty }$-category is a conglomerate of geometric shapes for higher structures with extra properties;

  • examples are: Kan complex, quasi-category, Segal category, opetopic n-category.

• in the algebraic definition of higher category an $\mathrm{\infty }$-category is a conglomerate of geometric shapes for higher structures with extra structure;

  • examples are: bicategory, tricategory, simplicially enriched category, strict omega-category.

One of the tasks of higher category theory is to relate and organize all these different models to a coherent general theory.

Strict versus weak

There are many different definitions of $\mathrm{\infty }$-categories, which may differ in particular in the degree to which certain structural identities are required to hold as equations or allowed to hold up to higher morphisms.

• See semi-strict infinity-category.

• See the discussion we had at discussion on terminology -- omega-category.

Literature

For a very gentle introduction to higher category theory, try The Tale of n-Categories, which begins in “week73” of This Week’s Finds and goes on from there… keep clicking the links.

For a slightly more formal but still pathetically easy introduction, try:

• John Baez, An Introduction to n-Categories, in 7th Conference on Category Theory and Computer Science, eds. E. Moggi and G. Rosolini, Springer Lecture Notes in Computer Science vol. 1290, Springer, Berlin, 1997.

For a free introductory text on $n$-categories that’s full of pictures, try this:

• Eugenia Cheng and Aaron Lauda, Higher-Dimensional Categories: An Illustrated Guidebook.

Tom Leinster has written about “comparative $\mathrm{\infty }$-categoriology” (to borrow a term):

• Tom Leinster, A Survey of Definitions of n-Category (arXiv)

• Tom Leinster, Higher Operads, Higher Categories (arXiv)

Recently $(\mathrm{\infty },1)$-categories (see homotopy theory) have attracted much attention :

• Jacob Lurie, Higher Topos Theory (arXiv)

There’s a lot more to add here, even if we restrict ourselves to very general texts. (More specialized stuff should go under more specialized subcategories!)