nLab connected category

Idea

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

So the skeletal connected groupoids are precisely the delooping groupoids of groups.

In homotopy theory, groupoids model exactly the homotopy 1-types and connected groupoids model the connected homotopy 1-types. For instance the fundamental groupoid of a connected topological space is a connected groupoid.

Every category $C$ induces a free groupoid $F(C)$ by freely inverting all its morphisms. A category is called connected if the groupoid $F(C)$ is.

Definition

A category $C$ is connected if it is inhabited and the following equivalent conditions hold:

• any two objects can be connected by a zigzag of morphisms.

• the $\infty$-groupoidification of the category (the Kan fibrant replacement of its nerve) is a connected ∞-groupoid.

• the geometric realization of its nerve is a connected topological space.

• the localization $C[C_1^{-1}]$ of $C$ at all its morphisms is a connected groupoid.

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.

Related concepts

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

• connected homotopy type

References

The notion of connected groupoids was originally defined in

• Heinrich Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Mathematische Annalen 96 1 (1927) 360-366 [doi:10.1007/BF01209171]

whence some authors also speak of Brandt groupoids.

For more see the references at groupoids.