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!

  • $\begingroup$ some motivation: Segal space seems to refer interchangeably to a simplicial space with extra conditions or bisimplicial set with extra conditions. This makes me think that the two models are at least weakly equivalent in every way that matters, and so there should be weak equivalences $$ |ssSET(S,T)| \sim sTOP(|S|,|T|) $$ (these are inner homs) for $S$, $T$ Segal spaces (of the bisimplicial flavor). Trying to produce this weak equivalence led me to the above question. $\endgroup$ Mar 25 '11 at 21:26

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