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Stream: theory: category theory

Topic: Pseudocategory of spans internal to a 2-category

Patrick Nicodemus (Sep 24 2023 at 01:43):

Let $\mathcal{C}$ be a 2-category.
Let $X$ be an object in $\mathcal{C}$.

I propose the following "Internal bicategory of spans in $X$".

The object of spans in $X$, if it exists, is the 2-limit of the diagram $X\leftarrow X \rightarrow X$. It is equipped with a certain span $d\leftarrow u\rightarrow c$ in the hom-category $\mathcal{C}(\mathcal{S}(X),X)$, which is universal among spans in $\mathcal{C}(Y,X)$ for variable $Y$.
Say that $X$ has pullbacks if $\mathcal{C}(Y,X)$ has pullbacks for all $Y$ and they are preserved under precomposition with 1-cells $Y'\to Y$.
I claim that if $X$ has pullbacks, then there is a pseudocategory internal to $\mathcal{C}$ whose objects are $X$ and whose arrows are $\mathcal{S}(X)$, composition is as it is in the bicategory of spans.

Checking the details on this is a pain to do and a pain to write down (pentagon identity will be annoying) so i'd like a reference. Does anyone know of this? Is this a known construction?

Patrick Nicodemus (Sep 24 2023 at 01:45):

Also acceptable is a slick construction of this that allows me to say "the pentagon identity holds because this is a special case of this other thing where the pentagon identity is known to hold"

Mike Shulman (Sep 24 2023 at 04:28):

the 2-limit of the diagram $X\leftarrow X\to X$

Do you mean the lax limit?

Mike Shulman (Sep 24 2023 at 04:29):

It should also be the power (= cotensor) of $X$ by the "walking span" category $(\cdot \leftarrow \cdot \to \cdot)$.

Mike Shulman (Sep 24 2023 at 04:34):

For a slick construction, the first thing that occurs to me is that $\rm Span$ is a 2-functor $\rm PB \to Dbl$ from the 2-category of categories with pullbacks to the 2-category of double categories. Now the assumption that $X\in \mathcal{C}$ has pullbacks means that its representable functor $\mathcal{C}^{\rm op} \to \rm Cat$ actually lands in $\rm PB$. Composing we get $\mathcal{C}^{\rm op} \to \rm PB \to Dbl$. The assumption that the object of spans exists means that the composite of this functor with both the "category of objects" and the "category of morphisms" functors $\rm Dbl\to Cat$ are representable. So now the Yoneda lemma should imply that the associativity and coherence etc. in $\rm Dbl$ can equivalently be expressed internally in $\mathcal{C}$.

Patrick Nicodemus (Sep 24 2023 at 05:06):

Mike Shulman said:

the 2-limit of the diagram $X\leftarrow X\to X$

Do you mean the lax limit?

I don't know what the terminology is. I mean it represents the 2-functor $\mathcal{C}^{\rm op}\to \mathbf{Cat}$ sending $Y$ to the category of spans in $\mathcal{C}(Y,X)$.

Patrick Nicodemus (Sep 24 2023 at 05:11):

I also think the yoneda lemma is the way to go here but i don't quite follow you with how the double category structure comes into it.

Patrick Nicodemus (Sep 24 2023 at 05:13):

The associativity and coherence you're talking about are internal to the double categories, not between them, right?

Patrick Nicodemus (Sep 24 2023 at 05:27):

ohh I might see what you're saying. It sounds like you're saying something like, try and generalize the theorem that
if $Hom(-,X)$ is valued in the category of groups rather than just sets, then by Yoneda $X$ must be a group obje (assuming C has the Cartesian products necessary to form $X\times X$).
Similarly here if $Hom(-, C)$ is valued in double categories, $C$ must itself be a double category somehow.

Patrick Nicodemus (Sep 24 2023 at 05:44):

Like, somehow there's some slightly more general finite limit theory version of Yoneda at play?
Handwavy definition because it's slightly difficult to phrase it right:
A 2-categorical finite limit theory consists of a finite number of 0-cells $X_1,\dots, X_n$ and a finite number of 1-cells $g_1,\dots, g_m$ and 2-cells $\kappa_1,\dots, \kappa_s$ where the domain and codomain of $g_{k+1}$ are permitted to be, inductively, any (formal) finite lax limit built from the $X_1,\dots,X_n$ and $g_1,\dots, g_k$.

One should have a 2-category of models of this theory in any 2-category, and lax 2-functors induce functors between the associated 2-categories of models. (In particular, pseudocategories are supposed to be models of a particular "algebraic theory" in this sense.)

Now let $\mathcal{C}$ be a 2-category, and say we have a functor $A$ from $\mathcal{C}^{\rm op}$ into the category of models of a finite lax limit theory. $A$ can always be viewed, equivalently, as a single model of the theory in the functor category $[\mathcal{C}^{\rm op},\mathbf{Cat}]$. Now we claim that if each object in the diagram $A$ is representable, then by Yoneda all 1 and 2 cells in the model are also representable, and this lifts to a model of the theory in $\mathcal{C}$.

It's important here that Yoneda preserves finite limits. Does 2-categorical Yoneda preserve finite lax limits? Is that important here?

Mike Shulman (Sep 24 2023 at 06:42):

Yes, that's the right idea, except that in the general stuff about finite 2-limit theories you don't ever want to use the word "lax". Lax has a specific meaning, it refers to having noninvertible mediating 2-cells.

For instance, a lax functor doesn't preserve composition even up to isomorphism, it just has specified 2-cells $F(g) \circ F(f) \Rightarrow F(g\circ f)$ satisfying coherence laws. Lax functors are useful for some things, but when treating 2-categories as categorifications of 1-categories you almost never want to talk about lax functors as they rarely preserve anything interesting and aren't even invariant under equivalence. Instead you want either strict 2-functors, which preserve composition strictly, or pseudo 2-functors, which preserve it up to isomorphism -- they are lax functors whose comparison 2-cells are invertible.

A lax limit, on the other hand, is more useful, but is only a special case of the general notion of weighted 2-limit. The lax limit of a diagram $D$ is a universal object $L$ equipped with projections $p_x : L \to D x$ along with for every $f:x\to y$ a 2-cell $D f \circ p_x \Rightarrow p_y$, satisfying coherence laws. If $D$ is the diagram $X \leftarrow X \to X$, you can see that this is the "object of spans" in $X$. Whereas the strict 2-limit of this diagram (which has equalities $D f \circ p_x = p_y$) would be isomorphic to $X$, and the pseudo 2-limit would be equivalent to $X$. But there is a general notion of weighted 2-limit which includes strict ones, pseudo ones, and lax ones, as well as other things, and (as in enriched category theory more generally) that's what one would use when formulating a notion of finite-limit theory.