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Stream: theory: category theory

Topic: adjunctions and the Chu construction

sarahzrf (May 19 2020 at 20:40):

so if you have an adjunction $F \dashv G$, then $(F / \mathrm{id}) \simeq (\mathrm{id} / G)$—is the Chu construction $\operatorname{Chu}(C, d)$ the same as that category when the adjunction is the contravariant adjunction $[-, d] \dashv [-, d]$?

Mike Shulman (May 20 2020 at 05:06):

Seems right.

sarahzrf (May 20 2020 at 05:19):

hmmmm :)

sarahzrf (May 20 2020 at 05:33):

so, i was thinking about points vs opens—like, should we think of topological spaces or locales as primary?—& i was wondering if maybe we cldn't somehow have a balanced view where neither is the full data of a space, since the adjunction fails to be fully faithful in either direction

sarahzrf (May 20 2020 at 05:34):

so i was thinking what if you defined a space as being both a set of points and a frame of opens, with an element-of relation, and then it occurred to me that schemes are kinda like that (but for regular functions instead of opens)

sarahzrf (May 20 2020 at 05:34):

and then all of that stuff suddenly really strongly reminded me of the concepts surveyed in "linear logic for constructive mathematics" of a proposition as determined in a balanced way by what it means to prove it and what it means to refute it

sarahzrf (May 20 2020 at 05:35):

and that's doing a chu construction—but i guess a chu construction can be seen as a special case of an adjunction's comma category, for a particular ubiquitous sort of adjunction

sarahzrf (May 20 2020 at 05:36):

so what if you, like, took the comma category of, say, the adjunction between spaces and locales?

sarahzrf (May 20 2020 at 05:41):

an object would be a pair $(L, X, f)$ of a locale L and a topological space X, equipped with a continuous function from X to the points of L

sarahzrf (May 20 2020 at 05:43):

equivalently, a frame homomorphism $O(L) \to O(X)$

Morgan Rogers (he/him) (May 20 2020 at 09:15):

This reminds me a little of a talk on 'biframes' that I saw last year, where someone was trying to give a point-free version of a space equipped with two (possibly interacting) topologies.

Morgan Rogers (he/him) (May 20 2020 at 09:19):

sarahzrf said:

so, i was thinking about points vs opens—like, should we think of topological spaces or locales as primary?—& i was wondering if maybe we cldn't somehow have a balanced view where neither is the full data of a space, since the adjunction fails to be fully faithful in either direction

imo points are overrated: if you have points that your topology can't see properly, it's probably just a bad choice of topology :stuck_out_tongue_wink:

Oscar Cunningham (May 20 2020 at 09:28):

This reminds me of something I talked about here. The category of topological spaces is total, which implies that every limit-preserving functor $\mathbf{Top}^\mathrm{op}\to\mathbf{Set}$ is representable. Meanwhile, (I think) the category of locales has the property that products distribute over colimits.

These two properties between them imply cartesian closedness, because the functor $X\mapsto ((X\times A)\to B)$ would be limit-preserving and hence representable. But sadly the two pieces of the puzzle have gifted to different categories; pointless topology has one and pointy topology has the other.

Soichiro Fujii (May 20 2020 at 09:50):

This reminds me of “topological systems” of Vickers (see his Topology via Logic, Chapter 5). I think that topological systems are more restrictive than objects of the comma category in that $X$ is not a general topological space, but a discrete one.

Morgan Rogers (he/him) (May 20 2020 at 10:03):

Oscar Cunningham said:

But sadly the two pieces of the puzzle have gifted to different categories; pointless topology has one and pointy topology has the other.

It might be a significant amount of work, (and I suspect it's unlikely to work) but one could check to what extent the construction @sarahzrf is describing remedies those issues by unifying them, or if instead it compounds them.

sarahzrf (May 20 2020 at 13:30):

Soichiro Fujii said:

This reminds me of “topological systems” of Vickers (see his Topology via Logic, Chapter 5). I think that topological systems are more restrictive than objects of the comma category in that $X$ is not a general topological space, but a discrete one.

haha, someone else pointed me to the same reference when i posted these thoughts on twitter :)

sarahzrf (May 20 2020 at 13:31):

and then i realized that @Mike Shulman was also citing that source to talk about the idea of a proposition in 2 parts...

sarahzrf (May 20 2020 at 13:31):

i should probably read that book.

Mike Shulman (May 20 2020 at 17:59):

It's a good book. And yes, I believe his topological systems embed in something like Chu(Poset,2). Although the really interesting question, I think, when embedding anything in a Chu construction, is what the $\ast$-autonomous structure of the Chu construction looks like on it.