end of a weak equivalence

2011-03-24T22:27:47Z

I would like to get a concrete description of sufficient conditions for the end of a morphism in $\mathcal{C}^{J^{op}\times J}$ (which is a point-wise weak equivalence) to be a weak equivalence.

In thinking about this problem, I've come to sufficient conditions that seem to be very rarely satisfied. Here's what I have:

An end is the same as a limit over the subdivison category, which I'll denote with $'$s. Subdivision categories are always inverse categories, and in particular Reedy, so we can put the Reedy model structure on $\mathcal{C}^{J'}$. When the index category is inverse, limit preserves trivial fibrations, so by Ken Brown's lemma, a limit of a point-wise weak equivalence between fibrant objects is a weak equivalence. So we need to figure out what fibrant means in $\mathcal{C}^{J'}$.

Let $X\in\mathcal{C}^{J^{op}\times J}$, and let $X'\in\mathcal{C}^{J'}$ be the associated 'subdivison'. If $f$ is a morphism of $J$, $M_f X' = \ast$ is the terminal object, because there are no non-identity morphisms with source $f$ in $J'$. That implies that in order for $X$ to be s.t. $X'$ is fibrant, for $f:s\rightarrow t$, we need $$ X(s,t) = X'(f) \rightarrow \ast\times_{M_f\ast}M_fX \cong \ast $$ to be a fibration. No surprises here--in order to be fibrant it needs to be point-wise fibrant (at least for objects that have a morphism between them).

When we look at objects $i\in J'$ coming from objects of $J$ though, the matching space becomes a limit over the discrete category of morphisms $f$ with source or target $i$. So the "matching space" condition here becomes $$ X(i,i) = X'(i) \rightarrow \ast\times_\ast M_i X' \cong \prod_{f:c\rightarrow i} X(c,i) \times \prod_{g:i\rightarrow c} X(i,c) $$ This seems like it's asking too much. A map into a product being a fibration would require something like the map to each factor being a fibration and all the lifts have to agree.

So, in short, the question is just:

Are there more reasonable sufficient conditions for the end of a weak-equivalence to be weak-equivalence?

Alternatively, if I made a mistake in the above, pointing it out would be great too!

end of a weak equivalence - MathOverflow

end of a weak equivalence