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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeMay 12th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI thought it was ridiculous that [[Span]] redirected to [[(infinity,n)-category of correspondences]], so I made a stubby page for it instead.

   I thought it was ridiculous that Span redirected to (infinity,n)-category of correspondences, so I made a stubby page for it instead.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMay 12th 2017
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. Added pointer to this page from the Examples-section at _[[2-category]]_.

   Thanks. Added pointer to this page from the Examples-section at 2-category.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeMay 12th 2017
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexGood. I think there is a lot more cross-linking that could be done here, and a lot more that could be added, but I don't have time right now.

   Good. I think there is a lot more cross-linking that could be done here, and a lot more that could be added, but I don’t have time right now.

4. • CommentRowNumber4.
   • CommentAuthormaxsnew
   • CommentTimeJun 28th 2018
   • (edited Jun 28th 2018)
   • PermaLink
   Author: maxsnew
   Format: MarkdownItexJust saw this interesting paper on the arxiv that gives a universal property for the bicategory of polynomials: https://arxiv.org/abs/1806.10477v1 . The intro goes over a universal property from Hermida ("Representable Multicategories") that really struck me as being about [[proarrow equipments]] so I thought I'd ask here. It says that a pseudofunctor out of Span(E) (as a bicategory) is the same thing as a "Beck pseudofunctor" out of E, which is a pseudofunctor out of E where every morphism has a right adjoint and satisfies a Beck-Chevalley condition. But it seems to me like we could slightly strengthen this result and make it more natural by noting that Span(E) is naturally an equipment (assume E has pullbacks) and in any map of equipments $F : C \to D$ a vertical morphism of $C$ gets sent to a vertical morphism in $D$ and thus an adjoint pair of horizontal morphisms in $D$. Then the very natural (if you like equipments) theorem would be that any pseudofunctor $DblSpan(E) \to C$ of equipments is equivalent to a functor $E \to Vert(C)$ where $Vert(C)$ is the vertical category of $C$ where $DblSpan$ is the double category of spans rather than the bicategory. Then we recover the original theorem by noting that the double category of adjunctions is a *right* adjoint, i.e., the maximal way to give a bicategory vertical arrows: a functor $C \to Adj(D)$ of equipments is the same as a pseudofunctor of bicategories $Hor(C) \to D$, where $Hor$ is the horizontal bicategory of $C$. So to sum it up it looks like we can decompose the functor $Span : CatPbk \to Bicat$ into two left adjoints: $$(DblSpan \dashv Vert) : CatPbk \to Equip$$ $$(Hor \dashv Adj) : Equip \to Bicat$$ so a pseudofunctor $Span(E) \to D$ is the same as $Hor(DblSpan(E)) \to D$ is the same as a functor $E \to Vert(Adj(D))$ which is the same as a Beck pseudofunctor. So that seems very nice to me: Span is the minimal way to give a category horizontal arrows and Adj is the maximal way to give a bicategory vertical arrows. If that's all true is there some way to make formulate Walker's result about polynomials in terms of triple categories? Out of my depth there because I don't actually know anything about triple categories but the formulation he uses involves a triple adjunction so maybe!

   Just saw this interesting paper on the arxiv that gives a universal property for the bicategory of polynomials: https://arxiv.org/abs/1806.10477v1 . The intro goes over a universal property from Hermida (“Representable Multicategories”) that really struck me as being about proarrow equipments so I thought I’d ask here.

   It says that a pseudofunctor out of Span(E) (as a bicategory) is the same thing as a “Beck pseudofunctor” out of E, which is a pseudofunctor out of E where every morphism has a right adjoint and satisfies a Beck-Chevalley condition. But it seems to me like we could slightly strengthen this result and make it more natural by noting that Span(E) is naturally an equipment (assume E has pullbacks) and in any map of equipments $F : C \to D$ a vertical morphism of $C$ gets sent to a vertical morphism in $D$ and thus an adjoint pair of horizontal morphisms in $D$. Then the very natural (if you like equipments) theorem would be that any pseudofunctor $DblSpan(E) \to C$ of equipments is equivalent to a functor $E \to Vert(C)$ where $Vert(C)$ is the vertical category of $C$ where $DblSpan$ is the double category of spans rather than the bicategory.

   Then we recover the original theorem by noting that the double category of adjunctions is a right adjoint, i.e., the maximal way to give a bicategory vertical arrows: a functor $C \to Adj(D)$ of equipments is the same as a pseudofunctor of bicategories $Hor(C) \to D$, where $Hor$ is the horizontal bicategory of $C$. So to sum it up it looks like we can decompose the functor $Span : CatPbk \to Bicat$ into two left adjoints:

   $$(DblSpan \dashv Vert) : CatPbk \to Equip$$ $$(Hor \dashv Adj) : Equip \to Bicat$$

   so a pseudofunctor $Span(E) \to D$ is the same as $Hor(DblSpan(E)) \to D$ is the same as a functor $E \to Vert(Adj(D))$ which is the same as a Beck pseudofunctor.

   So that seems very nice to me: Span is the minimal way to give a category horizontal arrows and Adj is the maximal way to give a bicategory vertical arrows. If that’s all true is there some way to make formulate Walker’s result about polynomials in terms of triple categories? Out of my depth there because I don’t actually know anything about triple categories but the formulation he uses involves a triple adjunction so maybe!

5. • CommentRowNumber5.
   • CommentAuthorMike Shulman
   • CommentTimeJun 28th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexYou seem to have dropped the crucial Beck-Chevalley condition?

   You seem to have dropped the crucial Beck-Chevalley condition?

6. • CommentRowNumber6.
   • CommentAuthormaxsnew
   • CommentTimeJun 28th 2018
   • (edited Jun 28th 2018)
   • PermaLink
   Author: maxsnew
   Format: MarkdownItexYou're right, but it looks like dropping the Beck-Chevalley condition is exactly what leads to a certain generalization of the theorem that uses normal oplax functors rather than pseudofunctors [1], which sounds even more like equipment stuff to me. I'll come back to this later because I am currently using it to procrastinate before a deadline :) [1] Dawson, Pare and Pronk, Universal Properties of Spans

   You’re right, but it looks like dropping the Beck-Chevalley condition is exactly what leads to a certain generalization of the theorem that uses normal oplax functors rather than pseudofunctors [1], which sounds even more like equipment stuff to me. I’ll come back to this later because I am currently using it to procrastinate before a deadline :)

   [1] Dawson, Pare and Pronk, Universal Properties of Spans

7. • CommentRowNumber7.
   • CommentAuthorMike Shulman
   • CommentTimeJun 28th 2018
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexNote that a sequel to that DDP paper, [The span construction](http://www.tac.mta.ca/tac/volumes/24/13/24-13abs.html), uses double categories explicitly.

   Note that a sequel to that DDP paper, The span construction, uses double categories explicitly.

8. • CommentRowNumber8.
   • CommentAuthormaxsnew
   • CommentTimeJun 28th 2018
   • PermaLink
   Author: maxsnew
   Format: MarkdownItexAh perfect, that's exactly what I was looking for: they go all the way and define Span to be an "oplax normal double category with all companions and conjoints" which might be also named a co-virtual equipment. So you get that Span gives you a co-virtual equipment and that it is the free co-virtual equipment wrt the forgetful functor that takes the vertical category and it is representable iff the original category has pullbacks. Would be very cool if polynomials turn out to be some kind of free virtualized triple category that's representable when the category is locally cartesian closed.

   Ah perfect, that’s exactly what I was looking for: they go all the way and define Span to be an “oplax normal double category with all companions and conjoints” which might be also named a co-virtual equipment.

   So you get that Span gives you a co-virtual equipment and that it is the free co-virtual equipment wrt the forgetful functor that takes the vertical category and it is representable iff the original category has pullbacks.

   Would be very cool if polynomials turn out to be some kind of free virtualized triple category that’s representable when the category is locally cartesian closed.

9. • CommentRowNumber9.
   • CommentAuthorvarkor
   • CommentTimeMay 26th 2022
   • PermaLink
   Author: varkor
   Format: MarkdownItexThis page currently has less information about the bicategory of spans than the page [[span]] does. Maybe some of the information on the latter could be moved to this page instead, or the two pages merged.

   This page currently has less information about the bicategory of spans than the page span does. Maybe some of the information on the latter could be moved to this page instead, or the two pages merged.