nLab
0-category

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Definition

A 00-category (or (0,0)(0,0)-category) is, up to equivalence, the same as a set (or class).

Remark

Although 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 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 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.

Remark

Interpreted literally, 00-category or (0,0)(0, 0)-category would be an \infty-category such that every jj-cell for j>0j \gt 0 is an equivalence, and any two such jj-cells that are parallel are equivalent. The picture that apparently emerges from this description might suggest a set equipped with an equivalence relation, or a what is sometimes called a setoid, or something even more complicated than that. One could thus say that a 00-category is a “setoid”, 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 00-category is equivalent to a set again.

Revised on July 2, 2014 00:04:26 by Toby Bartels (98.16.168.250)