Is there a category of topological spaces such that open surjections admit local sections? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T16:44:09Z http://mathoverflow.net/feeds/question/26576 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/26576/is-there-a-category-of-topological-spaces-such-that-open-surjections-admit-local Is there a category of topological spaces such that open surjections admit local sections? David Roberts 2010-05-31T13:40:27Z 2012-03-15T02:23:53Z <p>The class of open surjections $Q \to X$ is a Grothendieck pretopology on the category $Top$ of spaces, and includes the class of maps $\amalg U_\alpha \to X$ where ${U_\alpha}$ is an open cover of $X$. There are large classes of spaces for which these two pretopologies aren't equivalent (any non-locally contractible space, for example). What I'd like to know is if there are any spaces for which they are. My question can be split into two parts:</p> <ul> <li><p>Is there a full subcategory of $Top$ such that every open surjection admits local sections?</p></li> <li><p>Is there a non-full subcategory, like that of finite CW complexes and cellular maps (not that I'm claiming this is), in which every open surjection - in this category - admits local sections?</p></li> </ul> <p>Clearly the non-full subcategory part needs to include enough maps to be sensible, e.g. every continuous map is homotopic to one in the subcategory, and enough objects to also be considered nontrivial. </p> <hr> <p>Edit: I'm bumping this question because it received little interest, and I thought I'd explain the example which brought me to this idea.</p> <p>Consider the <em>path fibration</em> $P_xX \to X$, the total space of which is the space of based paths with the compact-open topology. If $X$ is locally contractible then this is fibre homotopy trivial, and in particular admits local sections. As $P_xX$ is contractible, it can be seen as a sort of 'free resolution' of the space $X$ - some sort of 'cover'. (If we work in the smooth setting, and let $X=G$ a Lie group, then $P_eG \to G$ is even a locally trivial $\Omega G$-bundle.) However, going to the other extreme and only asking for $X$ to be path connected and locally path connected then $P_xX \to X$ is an open surjection. There are lots of other maps which are open surjections which admit local sections (such as principal bundles)</p> <p>I'm happy for open surjections to be covers (i.e. form a Grothendieck pretopology) but then I want to be able to specify when open surjections and open covers generate the same Grothendieck topology, and this involves finding a category of spaces (which I'd like to be enough to model all homotopy types and mapping spaces correctly) where open surjections admit local sections.</p>