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Stream: learning: questions

Topic: ✔ Is the double category of spans closed?

James Deikun (Jan 25 2023 at 12:46):

Specifically, if $\mathcal{C}$ is a locally Cartesian closed category, does $\mathbb{S}\mathrm{pan}(\mathcal{C})$ have a right adjoint to span composition in each variable?

Bryce Clarke (Jan 25 2023 at 12:50):

James Deikun said:

Specifically, if $\mathcal{C}$ is a locally Cartesian closed category, does $\mathbb{S}\mathrm{pan}(\mathcal{C})$ have a right adjoint to span composition in each variable?

Yes! See Limit spaces and closed span categories by Brian Day.

Bryce Clarke (Jan 25 2023 at 12:50):

That result is for bicategories of spans, but I imagine that a similar result holds for the double category?

Notification Bot (Jan 25 2023 at 13:21):

Bryce Clarke has marked this topic as resolved.

Notification Bot (Jan 25 2023 at 16:10):

James Deikun has marked this topic as unresolved.

James Deikun (Jan 25 2023 at 16:11):

The reference calls this out as a "widely known fact" but does not provide a proof, cite a specific reference, or recall the construction of the adjoints.

Mike Shulman (Jan 25 2023 at 16:34):

Composing with a span $B \xleftarrow{f} A \xrightarrow{g} C$ is the composite functor $g_! f^*$. In an LCCC both of those functors have right adjoints, hence so does their composite: $f_* g^*$.

Mike Shulman (Jan 25 2023 at 16:36):

(As a side note, in general it seems to me that only the asker of a question should mark a topic as "resolved", just as on stackexchange only the asker of a question can mark an answer as "accepted".)

James Deikun (Jan 25 2023 at 23:56):

Oh, huh! I didn't think of using the composition functor in the underlying category to represent the arrow composition within span composition, that perspective is greatly clarifying... or so I thought until I tried to figure out what this did to the other arrow in the span :sweat_smile:

James Deikun (Jan 26 2023 at 08:06):

Specifically, if I have a span $E \xleftarrow{h} D \xrightarrow{k} C$ I can't seem to find an arrow from the domain of $f_{*}g^{*}k$ to $D$ or $E$. For example in Set when $f$ is not surjective, and $D$ and $E$ are empty sets, it seems like I need a map from a nonempty to an empty set.

Mike Shulman (Jan 26 2023 at 15:49):

Sorry, I was sloppy, I think it should be $(f\times 1_E)_*$.

James Deikun (Jan 28 2023 at 01:26):

Ah. It's tricky to see how $(f \times 1_E)^*$ does the right thing to a "compressed" span, but once you do everything else becomes obvious!

Notification Bot (Jan 28 2023 at 01:26):

James Deikun has marked this topic as resolved.