nLab inclusion function

Given a set XX and a subset AA of XX, the inclusion function of AA in XX is the function i A:AXi_A\colon A \to X given by

i A(n)=n i_A(n) = n

for every element nn of XX that belongs to AA.

Every inclusion function is an injection, and every injection is isomorphic (in the slice category Set/XSet/X) to an inclusion function. In appropriate categories, it is common to analogously interpret monomorphisms or regular monomorphisms as inclusions of subobjects. One then speaks of inclusion morphisms.

The inclusion function of AA in XX is the restriction to AA of the identity function on XX. Conversely, the restriction to AA of any function f:XYf\colon X \to Y is the composite of ff after the inclusion function:

i A=id X| A, i_A = {{\id_X}|_A} ,
f| A=fi A. {{f}|_A} = f \circ i_A .

Last revised on August 9, 2010 at 21:46:21. See the history of this page for a list of all contributions to it.