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Stream: deprecated: id my structure

Topic: Bicategory of split spans

Patrick Nicodemus (Aug 28 2023 at 01:58):

If $\mathcal{E}$ is a category with finite limits then there is a bicategory of "split spans". A 1-cell in the bicategory of split spans from $A$ to $B$ is an object $X$ equipped with $f : X\to A, g : X\to B$, $h : A\to X$ with $fh=1_A$.

Unit and Composition extends the unit and composition of spans. A 2-cell is a map of spans respecting the splitting.

Anybody see this before?

Similarly there should be a pseudo double category of $\mathcal{E}$ objects, $\mathcal{E}$ morphisms and split spans.

Intuitively if spans are generalized relations then these things are generalized total relations, they come with a function witnessing that the relation is total.

Patrick Nicodemus (Aug 28 2023 at 02:11):

Hm. Intuitively the property $ghf=g$ seems like it might be desirable for some arguments, but it doesn't appear to be necessary anywhere. On the other hand if you do want to assume this, it should be a sub-bicategory.
If spans are "bigraded sets" then spans satisfying this property are only trivially bigraded, as one grading is a refinement of another.

Patrick Nicodemus (Aug 28 2023 at 04:04):

Here's why I think this is an interesting notion. Consider the bicategory of sets and spans. Let $\mathcal{C}$ be a category and let $F : \mathcal{C}\to\mathbf{Span}$ be a lax functor. So $F$ associates to each $c$ a set $F(c)$ and to each $f : c\to d$ a family of sets $F(f)_{x,y}$ for $x\in F(c), y\in F(d)$. These relations are reflexive, so $F(1_c)(x,x)$ is true for every $x$, and transitive, so $F(f)(x,y) \land F(g)(y,z)\implies F(gf)(x,z)$.

If $f : c\to d$, then to prove a relation $F(f)$ is "total valued" is to give an element of $\prod_{x: F(c)}\sum_{y: F(d)}F(f)(x,y)$.
If we want to prove this for all $f$, there are natural coherence conditions that should be respected associated to the identity and composition of the category, and this entails that we lift the functor $F$ to the bicategory of split spans.

Patrick Nicodemus (Aug 28 2023 at 04:39):

For example, there is a functor $F : \mathbf{Cat}^{\rm op}\to \mathbf{Span}$ sending each $C$ to $\mathbf{Ob}(C)$ and $R : D\to C$ to $F(R)(c,d):= \left\{ \eta : c\to R(d)\mid \eta \textrm{ is a universal arrow} \right\}$. $R$ has a left adjoint iff $F$ is a total-valued relation, i.e., if there is an element of $\prod_{c}\sum_dF(R)(c,d)$. If we have a diagram in $\mathbf{Cat}^{\rm op}$, then we can lift it to the bicategory of split spans iff it is possible to choose a left adjoint for each functor in the diagram in a way that is compatible with composition on-the-nose.

As another example, let $p: \mathbb{E}\to \mathbb{B}$ be a Grothendieck fibration, and let $p^{-1} : \mathbb{B}^{\rm op}\to \mathbf{Span}$ be the lax functor which sends $b$ to $\mathbf{Ob}(p^{-1}(b))$ and for $f : c\to d$,$x\in p^{-1}(c), y\in p^{-1}(d)$, $p^{-1}(f)(x,y)$ is the set of Cartesian arrows from $x$ to $y$ over $f$.

Giving a lift of this lax functor to the bicategory of split spans is equivalent to choosing a splitting for the fibration.

Bryce Clarke (Aug 28 2023 at 04:47):

Hi @Patrick Nicodemus , I have studied this double category of "split spans" in Chapter 4 of my PhD thesis. There I called them "split multi-valued functions". It arises as the left-connected completion of the double category of spans.

Patrick Nicodemus (Aug 28 2023 at 04:48):

Sweet, thank you very much. Interesting! I'll check it out.

Bryce Clarke (Aug 28 2023 at 04:49):

The main goal of that work was construct a double category which "classifies" the notion of delta lens. Lax functors $F \colon B \rightarrow Span$ are the same as delta lenses $\int F \rightarrow B$.

Patrick Nicodemus (Aug 28 2023 at 04:51):

Alright, very nice, that will give me somewhere to start.

Patrick Nicodemus (Aug 28 2023 at 04:52):

Just for the sake of discussion, I am interested in studying the three dimensional version of this concept where the double category of spans is replaced with the intercategory of spans https://www.sciencedirect.com/science/article/abs/pii/S0022404916301256

Bryce Clarke (Aug 28 2023 at 04:53):

There are some talk slides (here, here, here) that I gave on this construction too.

Patrick Nicodemus (Aug 28 2023 at 04:54):

Oh, good. Thank you.

Patrick Nicodemus (Aug 28 2023 at 04:55):

I've alluded to this in previous threads but "lifting data" between morphisms of a category carries some kind of three dimensional structure

Bryce Clarke (Aug 28 2023 at 04:55):

Patrick Nicodemus said:

Just for the sake of discussion, I am interested in studying the three dimensional version of this concept where the double category of spans is replaced with the intercategory of spans https://www.sciencedirect.com/science/article/abs/pii/S0022404916301256

That sounds interesting! If you have something concrete you'd like to discuss, please feel welcome to contact me by email.

Patrick Nicodemus (Aug 28 2023 at 04:55):

Ok, great. I really appreciate that. Once i get a chance to write down some relatively clean initial thoughts I may send them along.

Tobias Fritz (Aug 28 2023 at 14:18):

Bryce Clarke said:

I have studied this double category of "split spans" in Chapter 4 of my [PhD thesis]

This sounds interesting, would you mind posting the thesis here? I currently have some difficulties with the download from behind the Chinese firewall, even with Tor.

(Background: taking a 1-categorical version of this construction gives a construction of Markov categories which reproduces some standard examples.)

Nathanael Arkor (Aug 28 2023 at 14:37):

Tobias Fritz said:

Bryce Clarke said:

I have studied this double category of "split spans" in Chapter 4 of my [PhD thesis]

This sounds interesting, would you mind posting the thesis here? I currently have some difficulties with the download from behind the Chinese firewall, even with Tor.

The-double-category-of-lenses.pdf