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Recall that the geometric realisation functor $| - |: sSet \to Top$ preserves products (choosing $Top = k Space$ or similar). Thus any given singleton Grothendieck pretopology on $Top$ gives rise to a singleton pretopology on $sSet$. I have in mind three, no four, examples:

  1. open covers (that is, $\coprod U_i \to X$ for $\{U_i\}$ an open cover of $X$)
  2. numerable open covers (ditto)
  3. open surjections
  4. maps admitting local sections (=locally split maps)

The pretopology $J^*$ on $sSet$ (which is a wide subcategory) arises as the pullback along $| - |$ of the given pretopology $J$ on $Top$. Now it occurs to one that there might be some sort of characterisation of maps of simplicial sets that correspond to, for example, open surjections. Since geometric realisation of maps in $sSet$ give rise to cellular maps (if we retain for argument's sake the cellular structure on the spaces involved), one could think of this as a characterisation of cellular maps which have the required property. But it seems to me that cellularity is sort of orthogonal (not the technical meaning!) to openness, in that it seems hard to think of a map of simplicial sets which is an open cover upon geometric realisation. Thus 1. and 2. don't seem very promising.

However, if one has a twisted cartesian product (see, e.g. May's book on simplicial sets), which is the 'same thing' as a fibre bundle in the world of simplicial sets, then geometric realisation gives you a fibre bundle (even a principal G-bundle), hence open surjection. But this is obviously too strong to give all of $J^*$ for either $J=$ open surjections or $J=$ locally split maps. So my question is:

Let $J$ be either the class of open surjections or the class of locally split maps in $Top$. Is a characterisation of those maps in $J^*$ likely? Possible? Already in the literature?

One could then compare this to the canonical topology on $sSet$, which consists of the epimorphisms. One would hope that $J^*$ is subcanonical, but this looks like it should follow from the definition...

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This not a complete answer but too long for a comment:

First a quick remark: 1.) and 4.) generate the same topology.

Now let geometric realization be denoted by $G$. Consider the geometric morphism $$\bar G=\left(G^*,G_*\right):Psh\left(Top\right) \to Psh\left(sSet\right),$$ and the geometric morphism $$J=\left(a,i\right):Sh_J\left(Top\right) \to Psh\left(Top\right)$$ corresponding to the Grothendieck topology $J$ (I have used $J$ to denote the geometric embedding since there is a one-to-one correspondence between geometric embeddings and Grothendieck topologies).

The topos of sheaves on $sSet$ with respect to what you call $J^*,$ is the pullback topos $$Sh_{J^*}\left(sSet\right):=Psh\left(sSet\right) \times_{Psh\left(Top\right)} Sh_J\left(Top\right).$$

Let $S:Top \to sSet$ denote the singular nerve functor. Note that $G^*$ is the left-Kan extension along Yoneda of the functor $$y_G:Top \to Psh\left(sSet\right),$$

$$y_G\left(T\right):X \mapsto Hom_{Top}\left(G\left(X\right),T\right) \cong Hom_{sSet}\left(X,S\left(T\right)\right),$$

i.e. for topological space $T,$ $$G^*\left(y\left(T\right)\right)=y\left(S\left(T\right)\right),$$ where $y$ denotes the Yoneda embedding.

Since all of the Grothendieck topologies you mentioned are extensive, it follows that that the $J^*$ sieves are the ones generated by the ones of the form

$$S\left(\underset{j} \coprod V_j\right) \to S\left(V\right).$$

I do not think it follows "by definition" that $J^*$ is subcanonical, but maybe I am missing something.

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P.S. David, I tried to send you a math email, but it bounced back. You can find mine on my website (listed on my profile here). Can you drop me a line with your email? Thx. – David Carchedi May 19 '11 at 15:17
Thanks, David, but I really am interested in the pretopology, not the topology. Hmm, I think geometric realisation preserves regular epimorphisms (since it preserves colimits and finite limits), but of course it doesn't a priori to reflect them. Perhaps I was a bit hasty in thinking that it was straightforward to claim so. – David Roberts May 20 '11 at 2:15

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