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Stream: theory: category theory

Topic: morphism of modules over 2-rigs

fosco (Apr 29 2021 at 11:46):

I have a problem being sure that I'm giving a certain definition right:

I want to specialise the definition of actegory to the case where the category $D$ has coproducts and each action map $a_C : D \to D$ preserves coproducts. I want to specialise this further and assume $C$ also has coproducts, interacting well with the monoidal structure on $C$ so that both $C$ and the category of coproduct-preserving endofunctors $[D,D]_\amalg$ are 2-rigs, to the effect that the action, transposed to a monoidal functor $C \to [D,D]_\amalg$, is a morphism of 2-rigs.

So far, so good. This should capture/categorify the notion of a module $D$ over the rig $C$.

But then I'd like to speak about a "morphism of bimodules" to be a functor $F : D \to E$ that categorifies the notion of morphism of bimodules, understood as categories with coproducts with an action as above; now, I can intuitively understand what I should ask, i.e. isomorphisms $F(c.d)\cong c.Fd$... and mutual compatibilities (F preserves coproducts; the action preserves coproducts in both arguments, etc.) but... how much coherence is this assignment supposed to satisfy?

Nathanael Arkor (Apr 29 2021 at 12:06):

Is this a specialisation of a more general structure with two interacting monoidal structures, e.g. a linearly distributive category? In this case, you could just take this as a definition, and then observe which coherence diagrams are automatic in your case.

Reid Barton (Apr 29 2021 at 12:07):

I'm not sure whether you really have bimodules or just modules, but preservation of coproducts is a property and not a structure, so the only coherences you need are the ones for a (bi)module.

Reid Barton (Apr 29 2021 at 12:11):

I'm not sure where to find them written down at the moment, but I think you just need one that says that the composition $F(c_1.c_2.d) \cong c_1.F(c_2.d) \cong c_1.c_2.F(d)$ agrees with (up to the associativity isomorphisms of $D$ and $E$) $F((c_1 \otimes c_2).d) \cong (c_1 \otimes c_2).Fd$ and likewise one that says $F(1.d) \cong 1.F(d)$ is the identity (up to the unit isomorphisms of $D$ and $E$).

Reid Barton (Apr 29 2021 at 12:16):

It's definition 4.1.7 of Hovey's book on model categories although the conditions are written out to about the same level of detail that I wrote them above.

fosco (Apr 29 2021 at 12:24):

Oh, thanks, I'll check Hovey

fosco (Apr 29 2021 at 13:07):

it seems a reasonable condition, also motivated by the use Hovey has in mind

fosco (Apr 29 2021 at 13:07):

...but now I feel obliged to think about examples of "model 2-rig" :grinning:

Joe Moeller (Apr 29 2021 at 13:13):

I think it's important to note that Hovey suppresses the word "closed" when talking about monoidal models categories.

Reid Barton (Apr 29 2021 at 14:33):

Right, so every module (or enriched) model category is an example.

Reid Barton (Apr 29 2021 at 14:34):

For my tastes it's "better" to categorify sums as cocompleteness or even local presentability, but maybe you have some other purpose in mind.

Reid Barton (Apr 29 2021 at 14:50):

Here is the framework that I like to use to understand these kinds of definitions. Fix a Cat-multicategory M. This is like a multi- version of a strict 2-category. It's just what you get by defining a V-multicategory for a (symmetric? to be safe) monoidal category V, and then setting V = Cat; there isn't any higher non-strictness involved. The point is that you actually have this strictness in typical examples, such as:

• M = categories, morphisms $(C_1, \ldots, C_n) \to D$ = functors $F : C_1 \times \cdots \times C_n \to D$ & natural transformations between such.
• M = categories, morphisms $(C_1, \ldots, C_n) \to D$ = adjunctions of $n$-variables & natural transformations between the left adjoints. We could also impose some (co)completeness or local presentability hypotheses on the objects. In the latter case we get something equivalent to:
• M = locally presentable categories, morphisms $(C_1, \ldots, C_n) \to D$ = functors which preserve colimits in each variable.
• M = categories with coproducts, morphisms $(C_1, \ldots, C_n) \to D$ = functors which preserve coproducts in each variable, for your notion.

Reid Barton (Apr 29 2021 at 14:52):

Then, in such an M, you can define a pseudomonoid object to be an object $C$ together with a multiplication map $\mu : (C, C) \to C$, with an associator which is an invertible morphism of $M((C, C, C), C)$ between two objects built from $\mu$, satisfying the pentagon equation; and also a unit $1 : () \to C$, with some more data and equations, so that if you specialize to the first example M = categories and multifunctors, you get the ordinary definition of a monoidal category $C$.

Reid Barton (Apr 29 2021 at 14:55):

And then you can define a (pseudo)module over such a $C$ to be an object $D$ together with an action map $\alpha : (C, D) \to D$ together with suitable coherence isomorphisms and equations, mimicking the ordinary definition of a category with an action by $C$. And then define monoidal functors, morphisms of modules, etc. again by mimicking the usual definitions.

Reid Barton (Apr 29 2021 at 14:57):

The point is that M encodes all the ambient conditions like that your categories should have coproducts and your (bi)functors should preserve them (separately in each variable), etc. By using the multicategory framework, you don't have to worry about whether there is actually a "tensor product" $C \otimes C$ which represents the right kind of bifunctors. Sometimes it won't exist and sometimes it will, but even when it does exist it will basically never be strictly associative and so on, so for this particular application it's much easier to encode everything using the multicategory anyways.

Mike Shulman (Apr 29 2021 at 16:21):

In fact, the 2-multicategory of multivariable adjunctions is not just a multicategory but a (2-)polycategory.

Nicolas Blanco (Apr 30 2021 at 13:47):

First a remark: if you believe in weak 2-multicategory then there should be a weak 2-multicategory $\mathbf{Dist}$ of categories and distributors (or profunctors or bimodules). Furthermore, $\mathbf{Cat}$ the 2-multicategory of functors is the sub-2-multicategory of $\mathbf{Dist}$ consisting of distributors representable in their codomain. $\mathbf{MAdj}$ the 2-multicategory of multivariable adjunctions is the sub-2-multicategory of $\mathbf{Dist}$ (and $\mathbf{Cat}$) consisting of distributors representable in each of their variable.
As mentioned by @Reid Barton a pseudomonoid in $\mathbf{Cat}$ is a monoidal category. A pseudomonoid in $\mathbf{Dist}$ is a promonoidal category while a pseudomonoid in $\mathbf{MAdj}$ is a monoidal biclosed category.
Now there is an alternative way of thinking of pseudomonoid in 2-multicategories.

By defining $\mathbf{1}$ to be the terminal multicategory that has one object $\ast$ and exactly one multimap for each arity $\mu^n \colon \ast^n \to \ast$, a pseudomonoid in $M$ is equivalently a pseudofunctor $\mathbf{1} \to M$.
$\mathbf{1}$ turns out to be the free multicategory containing a monoid which explains the above fact. We could try to define the free multicategory containing a monoid and a module over it.
As I am always bad to come up with notion let call this $\mathbf{Act}$. If I am correct, it should contain two objects $s,m$ and it should have exactly one multimap $\mu^n \colon s^n \to s$ and exactly one multimap $\alpha^n \colon s^n, m \to m$ for each arity $n$. A pseudofunctor out of this should be the same as a pair of a pseudomonoid and a pseudomodule over it.

One last comment: when specialising to the 2-multicategory $\mathbf{Cat}$, we get a pseudofunctor in $\mathbf{Cat}$. This triggers an alarm in me asking for a Grothendieck construction. And indeed there is a multicategorical Grothendieck construction - that in the case where you have tensors specialises to @Joe Moeller and @Christina Vasilakopoulou monoidal Grothendieck construction - that turns any pseudofunctor in $\mathbf{Cat}$ into an opfibration of multicategories. So we get that an opfibration over $\mathbf{1}$ is a monoidal category which is specified by the universal property of the tensor.

Now we can look at opfibration over $\mathbf{Act}$ and we get that it is a multicategory where the objects are split into two collections $S$ and $M$ such that $S$ has tensors (specified by their universal property) and it acts on $M$ in the following sense:
For any $(e,a) \in S \times M$ there is $\Phi(e,a) \in M$ with a multimap $\phi \colon e, a \to \Phi(e,a)$ such that any multimap $\Sigma, e, a \to b$ (where $\Sigma$ is a list of objects in $S$ and $b \in M$) uniquely factors through $\phi$ followed by a multimap $\Sigma, \Phi(e,a) \to b$.

I am not sure if this helps the intuition of anybody other than me though!

Nicolas Blanco (Apr 30 2021 at 13:55):

Also to add to @Mike Shulman comment, we can also consider pseudofunctor from the terminal polycategory $\mathbf{1}$ into a 2-polycategory. This specifies a Frobenius pseudomonoid inside our 2-polycategory. In the case of $\mathbf{MAdj}$ this gives a $\ast$-autonomous category.
I am not sure if we can get any insight on actegories from the polycategorical perspective.

Ben MacAdam (Apr 30 2021 at 18:37):

fosco said:

I have a problem being sure that I'm giving a certain definition right:

I want to specialise the definition of actegory to the case where the category $D$ has coproducts and each action map $a_C : D \to D$ preserves coproducts. I want to specialise this further and assume $C$ also has coproducts, interacting well with the monoidal structure on $C$ so that both $C$ and the category of coproduct-preserving endofunctors $[D,D]_\amalg$ are 2-rigs, to the effect that the action, transposed to a monoidal functor $C \to [D,D]_\amalg$, is a morphism of 2-rigs.

So far, so good. This should capture/categorify the notion of a module $D$ over the rig $C$.

But then I'd like to speak about a "morphism of bimodules" to be a functor $F : D \to E$ that categorifies the notion of morphism of bimodules, understood as categories with coproducts with an action as above; now, I can intuitively understand what I should ask, i.e. isomorphisms $F(c.d)\cong c.Fd$... and mutual compatibilities (F preserves coproducts; the action preserves coproducts in both arguments, etc.) but... how much coherence is this assignment supposed to satisfy?

I've been working with strict actegories (tangent categories), but it seems like the easiest way to work with these things is to take a pseudomonoid in the appropriate 2-category with 2-products, and then looking at algebras of $C \times (-)$.

Morgan Rogers (he/him) (Oct 10 2022 at 18:42):

I've just read @Nicolas Blanco 's message here. Sounds like some useful multicategorical insights into acts in actegories :innocent:

Nicolas Blanco (Oct 14 2022 at 14:53):

Thanks @Morgan Rogers (he/him) Let me expand a little bit on the monoid action internal to an actegory that I mentioned in a private message.
So there is this multicategory $\mathbf{Act}$ which is the free multicategory containing an internal monoid action - by which I mean an internal monoid and an action of it on an object. Any functor of multicategories $a \colon \mathbf{Act} \to \mathcal{M}$ defines an internal monoid action in $\mathcal{M}$. Any pseudofunctor into a 2-multicategory defines an internal pseudomonoid action in it. In particular any pseudofunctor into $\mathbf{Cat}$ defines an pseudomonoid in $\mathbf{Cat}$, i.e. a monoidal category, acting on a category: an actegory. Now under the Grothendieck construction this corresponds to an opfibration of multicategories $p \colon \mathcal{E} \to \mathbf{Act}$.
So an actegory is an opfibration of multicategories $p \colon \mathcal{E} \to \mathbf{Act}$. A monoid action internal to $\mathcal{E}$ is a functor $a \colon \mathbf{Act} \to \mathcal{E}$. But you also want to impose that the monoid part is in the monoidal category, the object in the category and the action of monoidal category and monoid are compatible. This should be enough to ask that $p\circ a = id_{\mathbf{Act}}$.
To sum up, a monoid object internal to an actegory $p \colon \mathcal{E} \to \mathbf{Act}$ is a section of $p$.
I am not sure what to do from there.

Btw, there seems to be a general pattern (akin to some microcosm principle) there:

• A monoidal category is an opfibration of multicategories $p \colon \mathcal{E} \to \mathbf{1}$ and a monoid internal to it is a section of it
• A functor of categories is an opfibration of categories $p \colon \mathcal{E} \to \mathbf{2}$ and a section of it is a object in the domain of the functor, together with a morphism in the codomain from the image of the object I think
• A *-autonomous category is a bifibration of polycategories $p \colon \mathcal{E} \to \mathbf{1}$ and a section of it is a Frobenius monoid

Nicolas Blanco (Oct 14 2022 at 14:59):

It feels like it might be related to Daniel Licata, @Mike Shulman and @Mitchell Riley paper "A Fibrational Framework for Substructural and Modal Logics" but I read it a while ago, so I am not sure.

Morgan Rogers (he/him) (Oct 14 2022 at 15:02):

I'm not familiar enough with multicategories to see clearly what this (op)fibrational presentation of actegories gets me. Let's discuss this in person when I next see you!