Pushout (category Theory) - Examples of Pushouts

Examples of Pushouts

Here are some examples of pushouts in familiar categories. Note that in each case, we are only providing a construction of an object in the isomorphism class of pushouts; as mentioned above, there may be other ways to construct it, but they are all equivalent.

1. Suppose that X, Y, and Z as above are sets, and that f : Z → X and g : Z → Y are set functions. The pushout of f and g is the disjoint union of X and Y, where elements sharing a common preimage (in Z) are identified, together with certain morphisms from X and Y.

2. The construction of adjunction spaces is an example of pushouts in the category of topological spaces. More precisely, if Z is a subspace of Y and g : Z → Y is the inclusion map we can "glue" Y to another space X along Z using an "attaching map" f : Z → X. The result is the adjunction space which is just the pushout of f and g. More generally, all identification spaces may be regarded as pushouts in this way.

3. A special case of the above is the wedge sum or one-point union; here we take X and Y to be pointed spaces and Z the one-point space. Then the pushout is, the space obtained by gluing the basepoint of X to the basepoint of Y.

4. In the category of abelian groups, pushouts can be thought of as "direct sum with gluing" in the same way we think of adjunction spaces as "disjoint union with gluing". The zero group is a subgroup of every group, so for any abelian groups A and B, we have homomorphisms

f : 0 → A

and

g : 0 → B.

The pushout of these maps is the direct sum of A and B. Generalizing to the case where f and g are arbitrary homomorphisms from a common domain Z, one obtains for the pushout a quotient group of the direct sum; namely, we mod out by the subgroup consisting of pairs (f(z),-g(z)). Thus we have "glued" along the images of Z under f and g. A similar trick yields the pushout in the category of R-modules for any ring R.

5. In the category of groups, the pushout is called the free product with amalgamation. It shows up in the Seifert-van Kampen theorem of algebraic topology (see below).