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Let $M_1\subset M_2$ and $K_1\subset K_2$ be inclusions of simplicial monoids. If there is a weak equivalence $f:M_2\to K_2$ that restricts to a weak equivalence $f|_{M_1}:M_1\to K_1$, does this induce a weak equivalence on the cokernels ${\mathrm{coker}}(M_1\to M_2)\to {\mathrm{coker}}(K_1\to K_2)$?

I know that homotopy colimits respects weak equivalence, but for the special case of the cokernel, can we just use the ordinary colimit?

Edit: Here is my actual situation. Let $A$ be a pointed simplicial subset of $X$. Consider the map $J(A)\to J(X)$ that sends $a\mapsto a^2$ where $J$ is James construction (i.e. the reduced free monoid construction). I trying to show that the group reflection ${\mathrm{coker}}(J(A)\to J(X))\to {\mathrm{coker}}(F(A)\to F(X))$ is a weak equivalence (or at least induces isomorphisms in homology). Here $F$ is Milnor's construction (i.e. the reduced free group construction).

The problem I'm facing is that map $a\mapsto a^2$ is not induced by a simplicial map $A\to X$. Otherwise, given a simplicial map $f:A\to X$, then ${\mathrm{coker}}(J(A)\xrightarrow{J(f)} J(X))= J({\mathrm{coker}}(A\xrightarrow{f} X))$ and we are done.

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What's your notion of weak equivalence? – David Roberts May 14 '12 at 5:50
The map induces isomoprhisms on homotopy groups of the simplicial monoids viewed as simplicial sets. – George May 14 '12 at 7:56
If all you want is a map which induces isomorphisms on $\pi_n$ for all $n$, then why can't you just use the long exact sequence associated to your map of cokernel sequences (i.e. Puppe Sequence)? You have $\cdots \pi_n(M_1)\rightarrow \pi_n(M_2) \rightarrow \pi_n(Cok) \rightarrow \pi_{n-1}(M_1) \rightarrow \pi_{n-1}(M_2)\cdots$ mapping down to a similar sequence with $K$ instead of $M$, so the 5-lemma tells you the middle vertical map (i.e. $\pi(Cok)\rightarrow \pi(Cok)$ must be an isomorphism. This holds for all $n$, with some fiddling for $n=0$ – David White May 14 '12 at 13:55
@David White: you seem to be assuming that $M_1\to M_2\to Cok$ is a fibration, which is true if $M_1$ is a group, but not in general. Cokernels for maps of monoids can be quite unpleasant. My guess is that the answer to the original question is negative. – Neil Strickland May 14 '12 at 14:35
@Neil Strickland: thanks for the correction. I was thinking $M_1\rightarrow M_2$ was the inclusion of a neighborhood deformation retract, but that wasn't in George's question, so I shouldn't have. I wonder if he can tell us more about this inclusion and how nice it is. There must be some application for this question which might shed more light on why this should or should not be true. – David White May 14 '12 at 16:39

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