# When does a pushout of monics induces a monic arrow?

There are lots of arguments in the homotopy theory of simplicial sets (I refer bascially to the first chapter of Goerss-Jardine) which exploit the fact that in certain cases the map induced in a pushout diagram made of monic arrows by two monic arrows is again monic.

The basic claim

BC: the map induced in a pushout diagram made of monic arrows by two monic arrows is again monic.

is false in general, even in "well-behaved" categories: consider the pushout diagram $$\begin{array}{ccc} \varnothing &\to & 1 \\ \downarrow &&\downarrow\\ 1 &\to &2 \end{array}$$ and the arrows $1\to 1$ on both sides.

So my question: I am looking either for conditions which ensure that BC is true or for a method to verify that a given arrow in a pushout diagram is/can't be a monic. I am in particular interested in the arrow naturally induced by $(K\times X)\coprod_{K\times Y}(L\times Y)\to L\times K$, by two monics $K\subset L, Y\subset X$. The product functor preserves monics, pushout of monics in a topos is monic, hence I'm left with a diagram $$\begin{array}{ccc} K\times Y &\to& L\times Y \\ \downarrow&&\downarrow\\ K\times X&\to&L\times X \end{array}$$ which must factor through the pushout. Is this factorization made of monics?

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But BC (in its full generality) is false in $\mathbf{sSet}$ as you have already observed! The pushout–product map you are interested in is a monomorphism because this particular pushout computes the union of subobjects. This is true in any elementary topos. –  Zhen Lin Aug 20 '13 at 23:27
Ah, now I see the light: $(L\times Y)\cap (K\times X) = K\times Y$, as in diagram 2. here ncatlab.org/nlab/show/subobject#LimitsAndColimits So simple that I feel ashamed. –  tetrapharmakon Aug 20 '13 at 23:35