Peter Selinger, a student here at Penn, observes that the category of topological spaces and local homeomorphisms fails to be complete only because it fails to have a terminator. That is, every _non-empty_ small diagram has a limit. Who knows a reference for this? Peter Freyd
In my last posting I had written: Peter Selinger, a student here at Penn, observes that the category of topological spaces and local homeomorphisms fails to be complete only because it fails to have a terminator. That is, every _non-empty_ small diagram has a limit. Who knows a reference for this? What I didn't realize is that he was correcting an error in "Categories, Allegories." On page 50 (1.461) we wrote that the category in question fails to have binary products. Wrong. I'm concluding that this was not known before. It's a nice example of a product that's very much not preserved by the forgetful functor. The product of the space of rational numbers with itself is uncountably large. (And for every infinite cardinal there's a space whose product with itself has the next power-cardinal as its size.)
Would the category of topological spaces and partial local homeomorphisms be complete? I mean, partial maps which are local homeomorphism on their locus of definition. David
Allow me please to muse on Peter Freyd's two postings concerning observations of Peter Selinger and himself on the category of topological spaces and local homeomorphisms. Is the product of X_1 and X_2 identical to the etale space of the sheaf that associates to an open set U of X_1 all the local homeomorphisms from U to X_2? Of course if this is right, then the roles of X_1 and X_2 can be reversed. Indeed, both etale spaces consist of germs of homeomorphisms between X_1 and X_2. The advantage of taking this point of view, besides that fact that perhaps the sheaf is easier to thing about than its etale space, is that it suggests persuing a more general construction. In a sense that I don't yet know how to make precise, X_1 x X_2 is a schizophrenic object because it lives in both toposes of sheaves, over X_1 and X_2, and it is in some sense the universal such schizophrenic object. So my question is, can this be made precise so that one could ask for a universal schizophrenic given any two toposes. My guess is that the analogous construction given the topos of M_1-sets and M_2-sets, M_1 and M_2 being monoids, would be to construct the free product M_1 * M_2, which also lives naturally in both categories. Moreover all M_1 * M_2-sets live in both categories, just as all sheaves over X_1 x X_2 induce sheaves over X_1 and over X_2. I don't know much about toposes. Is there already some well-known construction that fits the bill? David Feldman University of New Hampshire
Re: local homeomorphisms, and etendues Comment on Feldman's description of binary products in the category of spaces and local homeomorphisms. The product of an object X with itself is always the (object of arrows of) a groupoid, usually a rather uninteresting one. But in the category of spaces and local homeomorphisms, it is a very interesting groupoid, usually denoted \Gamma X (the pseudogroup of (germs of) local automorphisms of X, considered by Haefliger (and others?) back in the 1950's. It is also a groupoid in the category of spaces and all continuous maps, since pull-backs are preserved by the forgetful functor. For similar reasons, X times Y is a principal bundle over the groupoid \Gamma X (acting from the left), and \Gamma Y (acting from the right). Similar considerations apply in the category of locales. The corresponding localic groupoids were in essence considered by Ehresmann in "Gattungen von lokalen Strukturen", under the name of _local_ groupoids. I presented an expose about this, and its relatinship to etendue theory at the PSSL in Sussex in March, and the manuscript I circulated, may be picked up by ftp from theory.doc.ic.ac.uk, directory papers/Kock. It is called sussex.dvi. Anders Kock
participants (3)
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D_FELDMAN＠UNHH.UNH.EDU -
kock＠mi.aau.dk -
Peter Freyd