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How to define topology for general categories. The following is my attempt to do so but I guess it is not correct. What should be the categorical analouge of the third axiom of topology (that the union of open sets is open.)

A topology for an object $X \in C$ is a subcategory $\Omega$ of the category of subobjects of $X$, which:

  • contains the terminal object of $Sub\left(X\right)$, namely $id: X\rightarrow X $ and the empty suobject (the no morphism from the initial object of $C$ to X). This condition is generalised from of saying that $X$ and the empty set are open.

  • $\Omega$ is closed under finite pullbacks.(The pullback of any finite number of monics with codomain $X$ is included in $\Omega$). This must resemble the closure under intersection.

  • ??

Finally, I wanted to know this attempt is any close to the idea of sieve (or of Grothendiek's Topology ).

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It seems like a better idea to internalize the notion of locale (ncatlab.org/nlab/show/locale); IIRC the notion of topological space is known to be poorly behaved in the absence of choice (so in various topoi). –  Qiaochu Yuan May 3 '13 at 19:49
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There's an obvious way of categorifying the third axiom: say that the subcategory has all colimits. But this is still not enough. You should ask that finite meets distribute over all joins. In any case, one will not get close to the idea of a Grothendieck topology in this way... they are not really related to topologies at all. –  Zhen Lin May 3 '13 at 20:06
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Regarding Grothendieck topologies: a Grothendieck topology is a category with extra structure modelled on the idea that the open subsets of a topological space form a category (with the morphisms simply being inclusions, when they exist). The extra structure specifies which morphisms are to be considered covers (and so is related to the data contained in the "union" axiom of topology).

The reason for thinking about topologies in this way, and generalizing them, is that sheaves and presheaves on a topological space can be thought of as (contravariant) functors on the category of open subsets. (A presheaf is precisely such a functor, and a sheaf is a presheaf satisfying some additional axioms related to covers; these notions generalize to a Grothendieck topology essentially directly.)

What you are doing is more along the lines of trying to define topological objects in a category (analogous to topological spaces in the category of sets), which is something quite different.

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