Sheaves do not belong to algebraic geometry

Question

In the n-cafe post https://golem.ph.utexas.edu/category/2010/02/sheaves_do_not_belong_to_algeb.html Tom Leinster points out that sheaves on a topological space may be viewed in light of the nerve realization paradigm (the nerve realization adjunction induces an equivalence of Etalé spaces and sheaves). He also added a link to the latex file http://www.maths.ed.ac.uk/~tl/sheaves.pdf where in the very end of the document Leinster asks the reader to generalize what has been done with sheaves on a top. space to sheaves on a grothendieck topology. I am somewhat stumped there, is there even a way to generalize this to Grothendieck topologies? I don't yet see an analogue for the poset of open subsets $\text{Open}(S)$ of a top. space $S$. Does someone have any further insights on this? :)

Answer 1

I am somewhat stumped there, is there even a way to generalize this to Grothendieck topologies? I don't yet see an analogue for the poset of open subsets Open(S) of a top. space S.

The etale space construction for sites works in the same way as for topological spaces.

The etale space construction is a cocontinuous functor from sheaves to spaces, and cocontinuous functors from the category of sheaves of sets on a site $S$ can be identified with functors from $S$ such that covering sieves are mapped to isomorphisms.

If $S$ is the category of open subsets of a topological space $X$, then such a functor is given by sending $U\subset X$ to the map of topological spaces $U→X$.

If $S$ is an arbitrary site, one can supply such a functor manually. In particular, we can specify a covering of the topos of sheaves of sets on $S$ by a localic topos (i.e., the topos of sheaves on a locale $L$), and the inverse image functor $f^*$ sends an object $P∈S$ (interpreted as a map $P→1$) to an etale map of locales $f^*P→f^*1=L$, where sheaves of sets on $L$ are identified with their etale spaces.

Any Grothendieck topos admits a (nonunique) localic covering, so this construction always provides an etale space functor.