Dear Jean, I think I understood that, but my point was that if the structure functor fo= r a category of topological spaces was to have codomain Sets, given by space= |-> set of points, then these topological examples could be misleading. The= y are obviously different in nature from the algebraic examples (groups, lat= tices, etc.), and the idea that they at least have faithfulness in common is= a foundational accident. I should check that I understand your reference to toposes correctly, with f= unctors to CAT. Do you just mean the topos as category, in other words - if t= he topos is generalized space - the category of sheaves? That is actually a p= oint-free gadget. The correct analogy for topological spaces would then be t= hat the structure functor has codomain Poset^op, with space |-> lattice of o= pens. That is indeed faithful, but it is not the kind of "topological catego= ry" discussed in Joy of Cat. In fact it is very much of the algebraic nature= . That is why I think one should be suspicious of "commonly agreed" notions of= structure functor. They may be trying to encompass misleading examples. Regards, Steve.
On 12 Feb 2017, at 07:00, Jean Benabou <jean.benabou@wanadoo.fr> wrote: =20 =20 Dear Steve, My question was NOT what is a structure on an object of a category, but wh= at is a structure FUNCTOR p: S --> X. Topological spaces are the objects of= many DIFFERENT categories S, e.g. with morphisms continuous, open, closed, p= roper maps, each of these categories is equipped with a functor with codoma= in Sets. All these functors coincide on the objects, but their properties AS= FUNCTORS are completely different. The situation is the same with toposes w= ith geometric, open, closed, and proper morphisms and the structure functorS= with codomain in CAT. (As you see neither points nor whales are involved; b= ut they could be with Fred's help ). =20
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