Homotopy Type Theory
wild category

Wild categories


A wild category is a 1-dimensional approximation of an infinity-category that is definable in Book HoTT?. It consists of

If each A(x,y)A(x,y) is a set, then a wild category reduces to a precategory, but in general this condition is not imposed. This means that, for instance, there is a nontrivial pentagon identity for the associativities that does not necessarily commute, and so on. However, even lacking these coherence data, a wild category is sufficient for some purposes.

For example, we can define an initial object in a wild category to be an object 00 such that A(0,x)A(0,x) is contractible for all xx. In cases when the wild category “is” actually a coherent higher category, this still gives the right answer, and it is sufficient for applications such as producing induction principles for higher inductive types. See the references for more specific examples.