# Sufficient conditions for smooth pushout

We restrict ourselves to the category of smooth manifolds and smooth maps.

Suppose we have a pair of smooth maps $f:A \to X$ and $g:A \to Y$. A pushout is a pair of smooth maps $p:X \to Z$ and $q:Y \to Z$ satisfying $pf=qg$ and the following universal property:

($*$) For every pair of smooth maps $r:X \to W$ and $s:Y \to W$ satisfying $rf=sg$, there is a unique smooth map $u: Z \to W$ such that $up=r$ and $ur=s$.

Question: Are any (relatively convenient) conditions for the existence of pushout known?

In the case of pullback, it is known that we have a pullback if one of $f:X \to B$ and $g:Y \to B$ is a submersion (or $f$ and $g$ are transverse). I am asking about such a condition for pushout.

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I don't have an answer for you, but this might be a starting point. Every pushout can be expressed as a coequalizer of a coproduct. Since there shouldn't be any problems with the existence of coproducts (as long as you allow disconnected manifolds), it might be fruitful to look at existence conditions for coequalizers (which are often a bit easier to get a handle on than pushouts). – Unwisdom Feb 25 '14 at 16:37

It's $X\coprod_A Y \simeq X \coprod Y /\sim$ where $\forall a,b \in A, f(a) \sim g(b)$

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And why should this be a smooth manifold? – Zhen Lin Feb 25 '14 at 22:13
oops... one point for you! – Léo Feb 25 '14 at 22:17
The relevant area seems to be Godement's theorem on quotients of differentiable spaces, and for a general look, see work of J. Pradines on Diptychs. (do a web search) – Ronnie Brown Feb 26 '14 at 16:18