nLab
0-category

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Higher category theory

higher category theory

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Basic theorems

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1-categorical presentations

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Definition

A 00-category (or (0,0)(0,0)-category) is simply a set (or class).

This terminology may seem strange at first, it simply follows the logic of nn-categories (and (n,r)(n,r)-categories). To understand these, it is very helpful 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 11-category of 00-categories; this is the category of sets. Then a category enriched over this is a 11-category (more precisely, a locally small category). Furthermore, an enriched groupoid is a groupoid (or 11-groupoid), so a 00-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.

Revised on November 10, 2010 09:50:29 by Urs Schreiber (87.212.203.135)