Classification of Quasi-topoi - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:59:32Z http://mathoverflow.net/feeds/question/41305 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/41305/classification-of-quasi-topoi Classification of Quasi-topoi David Carchedi 2010-10-06T17:50:58Z 2011-06-10T12:25:19Z <p>(Grothendieck) topoi are left-exact reflective subcategories of a category of presheaves. An important class of quasi-topoi (see: <a href="http://ncatlab.org/nlab/show/quasitopos" rel="nofollow">http://ncatlab.org/nlab/show/quasitopos</a>) arise as the category of concrete sheaves on a concrete site. Concrete sheaves are those sheaves <code>$X$</code> such that the induced map <code>$Hom(C,X) \to Hom(\underline{C},\underline{X})$</code> is injective for all objects $C$, where $\underline{C}$ is the underlying set of $C$ and <code>$\underline{X}$</code> is the value of <code>$X$</code> on the terminal object. Concrete sheaves are a reflective subcategory of all sheaves. Concrete sheaves are a particularly nice example of a quasi-topos as the resulting quasi-topos is both complete and cocomplete. My question is, is there a way to represent quasi-topoi (or nice ones) as reflective subcategories of a Grothendieck topos (with some condition on the reflector)? (Of course, for this, you'd need the quasi-topos to be complete, since reflective subcategories of complete categories are again complete). More generally, is there some theorem saying that a category is a (possibly non-complete) quasi-topos if and only if it can be embedded into a topos such that the embedding has such-and-such property?</p> http://mathoverflow.net/questions/41305/classification-of-quasi-topoi/42441#42441 Answer by Mike Shulman for Classification of Quasi-topoi Mike Shulman 2010-10-16T23:17:06Z 2010-10-16T23:17:06Z <p>This is only a partial answer, and you may know it already since I mentioned it recently at the nForum, but for completeness, here it is. Theorem C2.2.13 in <em>Sketches of an Elephant</em> shows that the following are equivalent for a category C:</p> <ol> <li>C is the separated objects for a Lawvere-Tierney topology on a Grothendieck topos (hence reflective in that topos)</li> <li>C is the category of sheaves for one Grothendieck topology on a small category which are also separated for a second Grothendieck topology</li> <li>C is a locally small, cocomplete quasitopos with a strong-generating set</li> <li>C is locally presentable, locally cartesian closed, and every strong equivalence relation is effective</li> </ol> <p>A category of this sort is called a <em>Grothendieck quasitopos</em>. The third characterization seems most similar to what you're looking for. I doubt you can get away without some generating-set condition, since it seems very unlikely that the (complete, cocomplete, locally small) quasitopos of pseudotopological spaces (for example) can be reflectively embedded in a topos.</p> <p>What I don't know is whether one can put conditions directly on a reflective subcategory of a topos, analogous to left-exactness of the reflector, to guarantee that it is of this form. The reflector for separated objects preserves finite products and monics, but I have no idea whether that would be sufficient as a characterization.</p> http://mathoverflow.net/questions/41305/classification-of-quasi-topoi/67433#67433 Answer by Steve Lack for Classification of Quasi-topoi Steve Lack 2011-06-10T12:25:19Z 2011-06-10T12:25:19Z <p>This is really an answer to the question raised in Mike's reformulation of the question, but is too long for a comment and may be of interest.</p> <p>Richard Garner and I have considered when a reflective subcategory of a presheaf category has the form considered in condition 2 of his answer. It turns out that preservation of finite products and monomorphisms is not enough: to see this, consider the reflection of directed graphs into preorders, which preserves finite products and monomorphisms but is not of this form. </p> <p>In fact for a full reflective subcategory <em>E</em> of a presheaf category [<em>C</em>^{op},Set], the following conditions are equivalent:</p> <ol> <li>there is a topology <em>j</em> and a larger topology <em>k</em> for which <em>E</em> consists of the objects which are sheaves for <em>j</em> and separated for <em>k</em></li> <li>the reflection preserves finite products and monomorphisms and is semi-left-exact</li> <li>the reflection preserves monomorphisms and has stable units</li> </ol> <p>Here the notions of semi-left-exactness and stable units come from </p> <blockquote> <p>Cassidy, Hebert, Kelly, Reflective subcategories, localizations, and factorization systems, J. Austral. Math. Soc. Ser. A, 38:287-329, 1985.</p> </blockquote> <p>Let <em>R</em> be the reflection and <em>r</em> the unit of the reflection. Semi-left-exactness says that <em>R</em> preserves each pullback of a component <em>rX:X->RX</em> of the unit along a map <em>A->RX</em> with <em>A</em> in the subcategory.</p> <p>Stable units says the same thing, but without the requirement that <em>A</em> be in the subcategory. This turns out to be equivalent to <em>R</em> preserving all pullbacks over an object of the subcategory. </p>