## Is there a category of topological-like spaces that forms a topos?

The category of convergence spaces generalise topological spaces and form a quasi-topos, as topoi are allegedly nicer is there a nicer kind of topological-like space, the category of which forms a topos?

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Are simplicial sets topological enough for you? – Qiaochu Yuan Sep 15 at 5:02
Can you choose a reasonable topology on the category of topological spaces, and take the category of sheaves for that topology? – Theo Johnson-Freyd Sep 15 at 5:09
@Theo: Unless we do some universe expansion, the category of sheaves on a large site is in general only an (infinitary) pretopos and not an (elementary) topos. @Mozibur: Have you looked at [Johnstone's topological topos](ncatlab.org/nlab/show/…)? – Zhen Lin Sep 15 at 8:55
@Zhen lin: do you have a reference for turning an infinitary pretopos into a topos by universe expansion? – Mozibur Ullah Sep 24 at 0:14

Ronnie has already given the answer which immediately popped into my head when I saw the question. But I should sound a warning that the objects of the topological topos aren't exactly topological spaces, and indeed it seems likely that the only full subcategories of $Top$ that are toposes are fairly uninteresting for topology: things like the full subcategory of discrete spaces or the full subcategory of codiscrete spaces, or the degenerate topos which consists of only a terminal object.

We think of topological spaces as sets with extra structure, meaning that the underlying-set functor $\hom(1, -): Top \to Set$ is faithful, i.e., morphisms of $Top$ are functions satisfying some property. Suppose $T$ is a full subcategory of $Top$ whose terminal object is a 1-element space, denoted as above by $1$. Then $\hom(1, -): T \to Set$ is again faithful. This means that if $T$ were a topos, it would be a well-pointed topos. If moreover $T$ were a complete well-pointed (nondegenerate) topos, then it would be equivalent to $Set$.

We see here a certain tension here between nice categories with some "un-nice" objects (e.g., the topological topos, some of whose objects are not "nice" topological spaces), and categories of "nice objects" which fail to be maximally nice as categories (e.g., toposes). But sometimes one can split the difference; the convergence spaces of the OP seem about as good a compromise as any.

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 I just reread this, and realised this is the kind of answer that I was looking for. – Mozibur Ullah Jan 22 at 4:10

MR0531162 Johnstone, P. T. On a topological topos. Proc. London Math. Soc. (3) 38 (1979), no. 2, 237–271.

"The author defines a topos E which has the category of sequential spaces F as a reflective subcategory whose inclusion functor preserves the colimit diagram arising from any open cover, certain colimit diagrams arising from closed covers, and the equalizer diagram giving the quotient space for an equivalence relation which is sequentially closed. The author answers the questions which spring to mind for the topos theorist about E:...."

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