homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
A -category (or -category) is, up to equivalence, the same as a set (or class).
Although this terminology may seem strange at first, it simply follows the logic of -categories (and -categories). To understand these, it is very helpful to use negative thinking to see sets as the beginning of a sequence of concepts: sets, categories, 2-categories, 3-categories, etc. Doing so reveals patterns such as the periodic table; it also sheds light on the theory of homotopy groups and n-stuff.
For example, there should be a -category of -categories; this is the category of sets. Then a category enriched over this is a -category (more precisely, a locally small category). Furthermore, an enriched groupoid is a groupoid (or -groupoid), so a -category is the same as a 0-groupoid.
To some extent, one can continue to define a (-1)-category to be a truth value and a (-2)-category to be a triviality (that is, there is exactly one). These don't fit the pattern perfectly; but the concepts of (-1)-groupoid and (-2)-groupoid for them do work perfectly, as does the concept of 0-poset for a truth value.
Interpreted literally, -category or -category would be an -category such that every -cell for is an equivalence, and any two such -cells that are parallel are equivalent. The picture that apparently emerges from this description might suggest a symmetric proset, a set equipped with an equivalence relation, or something even more complicated than that. One could thus say that a -category is a symmetric proset, when considered just up to isomorphism. But it is more appropriate in higher category theory to consider these things up to equivalence rather than up to isomorphism; when we do this, a -category is equivalent to a set again.
Last revised on January 13, 2024 at 18:18:38. See the history of this page for a list of all contributions to it.