Generalising how in an ordinary category, one has morphisms going between objects, and in a 2-category, one has both morphisms (or 1-morphisms or 1-cells) between objects and 2-morphisms (or 2-cells) going between 1-morphisms, in an -category, there are k-morphisms going between -morphisms for all . (The -morphisms are the objects of the -category.)
There are two crucially different uses of the term:
If one speaks strictly only of the joint generalization of category and ∞-groupoid, hence of the notion of internal category in homotopy theory, then the “-”-prefix is to be read as in A-∞ algebra, E-∞ algebra, L-∞ algebra and, in fact, A-∞ category: in all these cases it means that the defining structural relations such as associativity of morphisms are taken to hold up to coherent higher homotopy, also called strong homotopy.
In a more encompassing view on higher category theory one may take the maximal “weakening” of structures as implicit and speak of just 2-category to mean a bicategory or rather a (∞,2)-category, of just 3-category to mean a tricategory or rather a (∞,3)-category, of just 4-category to mean a tetracategory or rather (∞,4)-category, and so on. With this counting then an “-category” is some limiting notion of -category. With this meaning one also often speaks of ω-categories.
This is hence a much more encompassing notion of -category than that of (∞,1)-category. It is also much harder to formalize. While there is by now a very good (∞,1)-category theory/homotopy theory of (∞,n)-categories for all , the limiting case where is currently still poorly understood. While there are several existing proposed definitions for what a single ω-category is, in the most general sense, there is no real understanding of the correct morphisms between them, hence of the correct (∞,1)-category of ω-categories. But this may of course change with time.
If all the -morphisms in an -category are equivalences in some suitable sense, we call the -category an ∞-groupoid. In this case we can think of the -morphisms for as “homotopies” and the -groupoid as a model for a homotopy type. By analogy, we can, if we wish, think of an arbitrary -category as a combinatorial model for a directed homotopy type.
There are many different definitions realizing the general idea of -category. Models for -categories usually fall into two classes:
One of the tasks of higher category theory is to relate and organize all these different models to a coherent general theory.
There are many different definitions of -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.
For a very gentle introduction to notions of higher categories, 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:
For a free introductory text on -categories that’s full of pictures, try this: