nLab
connected category

Contents

Idea

A groupoid is connected if it is inhabited and every object is connected by a morphism to every other object.

Every category CC induces a groupoid G(C)G(C) by freely inverting all its morphisms. A category is connected if the groupoid G(C)G(C) is.

Definition

A category CC is connected if it is inhabited and the following equivalent conditions hold

Note that the empty category is not connected. For other purposes, one can argue about whether the empty set should be called “connected” (see connected space), but for the applications of connected categories, the empty category should definitely not be called connected. In particular, a terminal object is not a connected limit.

A connected limit is a limit whose domain diagram category is connected.

Last revised on October 17, 2010 at 07:54:40. See the history of this page for a list of all contributions to it.