What's the easiest example of a morphism of topoi that is not from that of a site? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T08:55:28Z http://mathoverflow.net/feeds/question/79572 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79572/whats-the-easiest-example-of-a-morphism-of-topoi-that-is-not-from-that-of-a-site What's the easiest example of a morphism of topoi that is not from that of a site? Yuhao Huang 2011-10-31T01:24:10Z 2012-04-17T15:43:16Z <p>A topos is defined to be a category that's equivalent to the category of sheaves on a site. Morphisms between topoi is defined by a pair of adjoint functors that behave like pull-back/push-forward of sheaves. But I was told one of the cool thing about topos is that sometimes there are morphisms of topos that are not from morphisms of a site. When people talk about this they mention the word "crystalline"...</p> <p>But is there a toy example I can play around with? What's the easiest example of this?</p> http://mathoverflow.net/questions/79572/whats-the-easiest-example-of-a-morphism-of-topoi-that-is-not-from-that-of-a-site/79597#79597 Answer by James Borger for What's the easiest example of a morphism of topoi that is not from that of a site? James Borger 2011-10-31T08:01:35Z 2011-10-31T21:30:27Z <p>Let $X$ be a scheme. Let $S$ be the site of open subschemes with the Zariski topology, and let $S'$ be the site of open affine subschemes with the Zariski topology. Let $T$ and $T'$ be the associated toposes. Let <code>$f\colon T\to T'$</code> be the topos map where <code>$f^*(U)=U$</code> for any affine open subscheme $U$. Then <code>$f$</code> is an equivalence because open affines form a base for the topology. Let $g$ be its inverse. Then <code>$g^*$</code> does not restrict to a map of sites: If <code>$V$</code> is an open subscheme, then <code>$g^*(V)$</code> is the sheaf "represented by $V$" (i.e. it sends an affine open $U$ to <code>$\mathrm{Hom}_X(U,V)$</code>), but if $V$ is not affine, then <code>$g^*(V)$</code> is not represented by an object in $S'$.</p> http://mathoverflow.net/questions/79572/whats-the-easiest-example-of-a-morphism-of-topoi-that-is-not-from-that-of-a-site/94300#94300 Answer by Filippo Alberto Edoardo for What's the easiest example of a morphism of topoi that is not from that of a site? Filippo Alberto Edoardo 2012-04-17T15:43:16Z 2012-04-17T15:43:16Z <p>As for you word "crystalline" it comes from the Crystalline Topos. In the book "Notes on crystalline cohomology" by Berthelot and Ogus, they show that a morphism $X\to X'$ of schemes (over a fixed base $S$) induces a morphism of the associated crystalline topoi althugh there is no morphism of the corresponding sites. This is discussed in Section 5, page 5.1.</p>