This question already has an answer here:

What is the dual category of topological spaces $Top$?

I know that the order theoretic dual of a topological space is a closed set system rather than an open set system. However, this doesn't answer my question (I think) since it is just a transformation of the objects, and not of the morphisms.

I thought that maybe it would be the same as $Top$ but have morphisms involving closed sets instead of open sets somehow -- however the definition of a continuous function as one for which preimages of closed sets are closed is equivalent to the definition for which preimages of open sets are open, so this doesn't seem to lead anywhere unless $Top$ is self-dual, which seems unlikely.

EDIT: let's see if we can start with a simple example and generalize.

Consider two copies of the real line, one with the discrete topology (call it $A$), and one with the regular Euclidean/metric topology (call it $B$).

The space of continuous functions from $A$ to $B$ is simply the space of all real-valued functions on the real line, i.e. $Hom(A,B)=\{\text{real valued functions on the real line}\}$.

So now for $Top^{opp}$, the dual category to topological spaces, $$Hom^{opp}(B,A)=\{\text{relations whose "inverses" are real valued functions on the real line}\}?$$

(since the range of a function doesn't have to be its codomain).

This is a simple/special case of two objects and their morphisms in $Top^{opp}$, but already the morphisms seem likely to be pretty useless in general since they aren't even functions in general.

Still, now what confuses me is that it seems like this problem wouldn't be unique to defining the dual category of $Top$; any category whose morphisms aren't all invertible would seem to have the same problem. Thus it seems more likely that I am misinterpreting the definitions.

EDIT: I am looking for a more precise definition than given in the answer to this question:

What is the opposite category of $\operatorname{Top}$?

However, it seems that looking at that question (which I couldn't find before asking this one) that there is no simple answer to what the dual category of topological spaces $Top$ is, which perhaps is unsurprising considering how pathological the simple example given above seems to be.

Therefore I am going to vote to close this question as a duplicate.


marked as duplicate by Chill2Macht, Claude Leibovici, R_D, Charles, Henno Brandsma general-topology Jun 18 '16 at 4:44

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ It should be reversing arrows, to be dual, right? $\endgroup$ – Henno Brandsma Jun 17 '16 at 13:07
  • $\begingroup$ Yes that is the definition of dual category $\endgroup$ – Chill2Macht Jun 17 '16 at 13:08
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    $\begingroup$ My (fairly uneducated) guess would be some kind of algebras, considering results such as Stone duality and Gelfand-Naimark duality. $\endgroup$ – mrp Jun 17 '16 at 13:54
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    $\begingroup$ Are you asking for a useful way to think about the opposite category, or for just what its precise definition is? For the former, see math.stackexchange.com/questions/1711330/…. $\endgroup$ – Eric Wofsey Jun 18 '16 at 2:11
  • $\begingroup$ More the latter. Still, I think this question is similar enough to the one you linked to (and which I should have searched harder to find before asking this question). So I will vote to close. $\endgroup$ – Chill2Macht Jun 18 '16 at 2:20

My first impulse would be to say frames, which are special lattices and where a continuous map induces a frame homomorphism in the opposite direction.

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    $\begingroup$ It isn't. The category of locales is not equivalent to the category of topological spaces. $\endgroup$ – Zhen Lin Jun 17 '16 at 13:21
  • $\begingroup$ @ZhenLin You mean that the dual of locales is frames and not Top, I see. $\endgroup$ – Henno Brandsma Jun 17 '16 at 13:34

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