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Stream: learning: questions

Topic: Change of ambient category

Keith Elliott Peterson (Jan 16 2024 at 10:54):

I'm just doing some internal category theory, and was wondering if there is such a thing as a change of basis of the underlying ambient categories, that is to say, turn $\textrm{Cat}(\mathcal{V})\text{s}$ into $\textrm{Cat}(\mathcal{W})\text{s}$ . I can't seem to find any resources on such.

Morgan Rogers (he/him) (Jan 16 2024 at 11:10):

The fact that pullback preserving functors V -> W induce functors between categories of internal categories does sometimes get handwaved. There is some stuff on internal categories in Part B of Sketches of an Elephant, iirc, so maybe you'll find the background you need there.

Damiano Mazza (Jan 16 2024 at 12:20):

Elaborating on what Morgan said, and if I understand your question correctly, the theory of categories $\mathbb{Cat}$ is a finite-limit theory (aka essentially algebraic theory, aka cartesian theory in the Elephant), so a category internal to a category with finite limits $\mathcal V$ is the same thing as a finite-limit-preserving functor $\mathrm{Syn}(\mathbb Cat)\to\mathcal V$, where $\mathrm{Syn}(\mathbb Cat)$ is the syntactic category of $\mathbb Cat$. Thus, if $\mathcal W$ is another category with finite limits, any finite-limit-preserving functor $\mathcal V\to\mathcal W$ sends a category internal to $\mathcal V$ to a category internal to $\mathcal W$, by postcomposition. However, this is too general to say anything specific about internal categories, so maybe the part of the Elephant that Morgan pointed to contains more specific remarks.

Evan Patterson (Jan 16 2024 at 18:25):

Here's a slick way to see it. Since spans compose by pullback, it is pretty direct to see that a pullback-preserving functor $F: \mathcal{V} \to \mathcal{W}$ induces a (pseudo) double functor $\mathbb{S}\mathsf{pan}(\mathcal{V}) \to \mathbb{S}\mathsf{pan}(\mathcal{W})$. Then, since an internal category in $\mathcal{V}$ is a lax double functor $\mathbb{1} \to \mathbb{S}\mathsf{pan}(\mathcal{V})$, you just need to post-compose to change the ambient category from $\mathcal{V}$ to $\mathcal{W}$.

Keith Elliott Peterson (Jan 19 2024 at 06:58):

Evan Patterson said:

Then, since an internal category in $\mathcal{V}$ is a lax double functor $\mathbb{1} \to \mathbb{S}\mathsf{pan}(\mathcal{V})$, you just need to post-compose to change the ambient category from $\mathcal{V}$ to $\mathcal{W}$.

Where is this result from?

John Baez (Jan 19 2024 at 07:39):

I don't know, but Evan will.

This result is connected to two perhaps more well-known results:

1. A monad in a bicategory $\mathbf{B}$ is the same as a lax monoidal functor $1 \to \mathbf{B}$.

2. A category is a monad in the bicategory $\mathbf{Span}(\mathsf{Set})$.

John Baez (Jan 19 2024 at 07:42):

But it would follow from two related claims - maybe someone can let me know if these are true or whether I'm just confused:

1'. A monad in a double category $\mathbb{D}$ is the same as a lax double functor $1 \to \mathbb{D}$.
2'. A category internal to a category $\mathsf{V}$ with finite limits is the same as a monad in the double category $\mathbb{S}\mathbf{pan}(\mathsf{V})$.

John Baez (Jan 19 2024 at 07:53):

Okay, 2'. is Example 2.6 in here:

• Thomas M. Fiore, Nicola Gambino and Joachim Kock, Monads in double categories.

and I wouldn't be shocked if 1'. were also in there.

Kevin Arlin (Jan 19 2024 at 17:56):

I think you’ll also find this all explained in Evan’s recent paper with Lambert, which is all about lax functors into spans giving you cool models of things. https://arxiv.org/abs/2310.05384

John Onstead (Jan 20 2024 at 08:25):

One thing I've always wondered about is how to construct the bicategory Span(V), for V a category of finite limits (or more generally with pullbacks), using purely category theoretic constructions. That is, how can I represent Span(V) using universal properties, limits, adjunctions, etc. with respect to V? And also importantly, can this construction generalize beyond Cat into DblCat so that we can construct double categories of spans? I believe I saw somewhere that the subcategory of Prof on discrete categories gives Span(Set), but this approach does not seem very generalizable, though maybe there is a way of generalizing it?

Nathanael Arkor (Jan 20 2024 at 09:13):

John Onstead said:

One thing I've always wondered about is how to construct the bicategory Span(V), for V a category of finite limits (or more generally with pullbacks), using purely category theoretic constructions. That is, how can I represent Span(V) using universal properties, limits, adjunctions, etc. with respect to V? And also importantly, can this construction generalize beyond Cat into DblCat so that we can construct double categories of spans? I believe I saw somewhere that the subcategory of Prof on discrete categories gives Span(Set), but this approach does not seem very generalizable, though maybe there is a way of generalizing it?

See Dawson–Paré–Pronk's two papers on this topic:

• Universal properties of Span
• The span construction

David Kern (Jan 21 2024 at 03:01):

Categories of spans are especially nice in that they have two universal properties, one for mapping out of — this is the one mentioned by by Nathanael Arkor — and one for mapping into (which is the one more useful for constructing monads in spans). The latter says that a functor from a (not double!) category $C$ into $\mathbb{Span}(V)$ is the same thing as a functor $F$ from the twisted arrow category $\mathsf{Tw}(C)$ (the category whose objects are arrows of $C$, and morphisms are factorisations of arrows) into $V$ satisfying the condition that for any composable triple $x\xrightarrow{\varphi}y\xrightarrow{\psi}z$ in $C$ we have $F(\psi\circ\varphi)\simeq F(\psi)\times_{F(1_y)}F(\varphi)$.
Because of this condition this doesn't quite express $\mathbb{Span}(-)$ as a right-adjoint — or relative right-adjoint I guess, if you want $\mathbb{Span}(V)$ as a double category and not just its horizontal/loose category — but it can be refined into one if you see $\mathsf{Tw}(C)$ (and also $V$) as what's called a “category with directions” or “adequate triple” (because their morphisms are the functors having this pullback condition).

Nathanael Arkor (Jan 21 2024 at 08:31):

David Kern said:

Because of this condition this doesn't quite express $\mathbb{Span}(-)$ as a right-adjoint — or relative right-adjoint I guess, if you want $\mathbb{Span}(V)$ as a double category and not just its horizontal/loose category — but it can be refined into one if you see $\mathsf{Tw}(C)$ (and also $V$) as what's called a “category with directions” or “adequate triple” (because their morphisms are the functors having this pullback condition).

Is there a reference for this?

David Kern (Jan 21 2024 at 18:21):

I unfortunately do not have any 1-categorical reference for this, only $\infty$-categorical (it was certainly known before for 1-categories, but I do not know that it was written down anywhere, possibly because 1-categories of spans are easy enough to manipulate by hand that looking that deeply into things wasn't as necessary). So going to what I do know, adequate triples and their span $\infty$-categories seem to have first appeared in Barwick's paper on spectral Mackey functors (under the name of effective Burnside $\infty$-category, in Definition 5.10), and the adjunction itself was worked out in an appendix of Sam Raskin's PhD thesis (section 20). I would say that a more “definitive” reference is Haugseng–Hebestreit–Linskens–Nuiten's recent paper on fibrations and spans (section 2), redoing everything in a much clearer and precise way (they only write it for the $(\infty,1)$-categories of spans, but I think it's easy enough to see that their arguments generalise straightforwardly to the double $\infty$-categories of spans).

Nathanael Arkor (Jan 21 2024 at 18:47):

Ah, I see, thank you!