Draft:homotopy quotient

Precisely, this is the homotopy colimit of the action groupoid.

It is also known as the Borel construction.

Quotient groupoid

Let

: $X \times G \,\overset{s}\underset{t}\rightrightarrows\, X$

be an action groupoid (usual set-theoretic). The colimit of the above diagram in Set is the coequalizer of it; i.e., the usual set-theoretic quotient.

Draft:homotopy quotient

Quotient groupoid

Further reading