Contents

Contents

Idea

An absolute pushout is a pushout which is preserved by any functor whatsoever. In general this happens because the pushout is a pushout for purely “diagrammatic” reasons. See absolute colimit for more.

Definitions

Definition

A particular pushout diagram in a particular category $C$ is an absolute pushout if it is preserved by every functor with domain $C$.

Equivalently, since the Yoneda embedding is the free cocompletion of $C$:

Definition

A particular pushout diagram in a particular category $C$ is an absolute pushout if it is preserved by the Yoneda embedding $C \hookrightarrow [C^{op},Set]$.

Split pushouts

We propose the following notion of split pushout.

Definition

A commutative square

defines a split pushout if there exist sections $p s = 1$, $q t = 1$, $m u = 1$

so that $p t = u n$.

Split pushouts are absolute pushouts

Proposition

Split pushouts are absolute pushouts.

Proof

Note that split pushouts are preserved by arbitrary functors, so it suffices to show that a split pushout is a pushout in the category in which it lives. To that end consider, a cone under the span $(p,q)$:

Upon composing with the sections to $q$ and $m$

we see that $c$ factors through the claimed pushout $P$ as $c = (b u) n$. We must verify that $b$ also factors as $b = (b u) m$. Since $p$ is an epimorphism, it suffices to prove that $b p = b u m p$, which follows easily:

$b p = c q = b u n q = b u m p.$

This produces the desired factorization. Finally, since $m$ is an epimorphism, such factorizations are unique.

Note the proof that a split pushout defines a pushout square in the category in which it lives did not require $p$ to be a split epimorphism. However, arbitrary functors do not preserve epimorphisms. They do however preserve split epimorphisms, and thus the section guarantees that the image of $p$ will define an epimorphism in any category.

Examples

In their study of generalized Reedy categories, Berger and Moerdijk introduce the notion of an Eilenberg-Zilber category, one of the axioms of which demands that spans of split epimorphisms admit absolute pushouts. In practice, this seems to be the case because the pushout of these split epimorphisms is a split epimorphism as above, often with an additional section $v$ of $n$ satisfying the additional equation that $v m = q s$.

The Berger-Moerdijk definition of an Eilenberg-Zilber category appears in: