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Stream: theory: category theory

Topic: V-enriched categories vs. categories with V-action

Tim Campion (Apr 22 2022 at 01:10):

It's well-known that if $V$ is a monoidal biclosed category, then a category $C$ with an action of $V$ which is cocontinuous in the $V$ variable is the same data as a $V$-enriched category $C$ which is tensored over $V$. So in my head, a category with an action of a monoidal category $V$ is a special case of a $V$-enriched category.

As pointed out to me recently by Tyler Lawson, it's important that this breaks down when $V$ is monoidal but not closed. For instance, if $V$ is monoids under cartesian product, then $V$ acts on itself via the monoidal product, but this does not correspond to a self-enrichment. And if you start thinking about how close you can get to talking about an internal hom, you end up thinking about centralizers -- a sort of new phenomenon which is kind of invisible in the monoidal biclosed context.

(More generally, $E_n$-algebras in some category $C$ admit an important monoidal structure which is not biclosed.)

So... I guess my question is: what are some other examples of monoidal categories $V$ which are not biclosed, and how does the notion of a category with $V$-action differ from, and relate to, the notion of a $V$-enriched category?

Mike Shulman (Apr 22 2022 at 03:54):

In the closed case, I think you need not just cocontinuity but that the action has a right adjoint. Of course if the categories are nice enough that follows from the adjoint functor theorem.

Mike Shulman (Apr 22 2022 at 04:01):

In fact I don't think closedness of $V$ has much to do with it. If the action has a right adjoint, it should enrich $C$ over $V$ regardless of whether $V$ is closed in its own right; and if the action doesn't have a right adjoint, it doesn't help to know that $V$ is closed.

Mike Shulman (Apr 22 2022 at 04:04):

For instance, if $G$ is a group, regarded as a discrete monoidal category, then it is biclosed with internal-homs $g^{-1}h$ and $h g^{-1}$. But a discrete $G$-actegory is just a $G$-set, while it is $G$-enriched only if it is a pseudo-torsor (i.e. the action is free and transitive).

Tim Campion (Apr 22 2022 at 06:18):

Oh that's food for thought... I was happy at first to assume that $V$ was locally presentable, but this is an interesting example where that doesn't hold!

Matteo Capucci (he/him) (Apr 22 2022 at 07:39):

The paper you want is 'A note on actions of monoidal categories' by Janelidze and (iirc) Kelly

Matteo Capucci (he/him) (Apr 22 2022 at 07:41):

V only needs to be symmetric monoidal for the theorem to work. Tensored V-enrichment is equivalent to a closed V-action, i.e. an action with a parameterised right adjoint (-*c: V -> C has a right adjoint for every c)

Tim Campion (Apr 22 2022 at 08:03):

Hm... is there some kind of adjunction between $V$-categories and $V$-actegories? Maybe with certain assumptions on $V$ which fall short of stipulating that it be biclosed, or that the tensor product be separately cocontinuous?

Matteo Capucci (he/him) (Apr 22 2022 at 08:23):

Tim Campion said:

Hm... is there some kind of adjunction between $V$-categories and $V$-actegories?

I had the same thought, and I tried really hard to prove it with @Dylan Braithwaite to no avail.

Matteo Capucci (he/him) (Apr 22 2022 at 08:23):

I thought it was easy and then it wasn't

Matteo Capucci (he/him) (Apr 22 2022 at 08:24):

Part of the difficulty is the difference between enrichment as structure and enrichment as property, which prompted my question at #theory: category theory > enrichment as a structure. It's much easier to generate free-forgetful adjunctions for structure than for property!

Matteo Capucci (he/him) (Apr 22 2022 at 08:28):

IIRC we actually did find an adjunction between 'V-structures' (i.e. V-valued profunctors $C^\circ \times C \to V$ for an existing category C) and V-actegories whose nucleus is the Janelidze-Kelly equivalence, but 'a category with V-structure' is not what most people call a 'V-category'

Dylan McDermott (Apr 22 2022 at 10:12):

Matteo Capucci (he/him) said:

V only needs to be symmetric monoidal for the theorem to work

The usual definition of "tensored" (e.g. in Kelly's book) needs $V$ to be closed, because it asks for isomorphisms

$$C(v * c, c') \cong v \multimap C(c, c')$$

where $C(c, c')$ is the hom-object, and $\multimap$ is the closed structure of $V$.

But $V$ doesn't need to be closed if you instead ask for natural bijections

$$V(v', C(v * c, c')) \cong V(v' \otimes v, C(c, c'))$$

(Symmetry isn't needed at all for this.)

Dylan McDermott (Apr 22 2022 at 10:19):

Tim Campion said:

Hm... is there some kind of adjunction between $V$-categories and $V$-actegories?

The way I prefer to think about this is that all of these things embed into Wood's "large V-categories" (which modulo size issues are categories enriched over $[V^{\mathrm{op}}, \mathbf{Set}]$ with Day convolution). Wood shows that there is a fully faithful 2-functor from $V$-categories to $[V^{\mathrm{op}}, \mathbf{Set}]$-categories, and the image of this is the $[V^{\mathrm{op}}, \mathbf{Set}]$-categories $C$ for which $C(c, c') : V^{\mathrm{op}} \to \mathbf{Set}$ is representable for all $c, c'$ (the representations are the hom-objects). There is a similar fully faithful 2-functor from the 2-category of left $\mathrm{V}$-actegories, strong functors and strong natural transformations, and the image is those $C$ for which $C(c, {-})v : C_0 \to \mathbf{Set}$ is representable. ($C_0$ is the underlying ordinary category.)

When $C(c, {-})v$ and $C(c, c')$ are representable (so there is both an enrichment and an actegory), the representations form an adjunction, and this is one way of proving the theorem about closed actegories and tensored $V$-categories.

Alexander Campbell (Apr 22 2022 at 10:42):

@Tim Campion I give a treatment of this comparison from the "enrichment as structure" POV in Section 4 of my paper on Skew-enriched categories. The idea is that you can see V-categories and V-actegories as special kinds of skew V-proactegories. (N.B. To capture V-categories you really do need to consider skew V-proactegories!)

Matteo Capucci (he/him) (Apr 22 2022 at 13:00):

Dylan McDermott said:

Matteo Capucci (he/him) said:

V only needs to be symmetric monoidal for the theorem to work

The usual definition of "tensored" (e.g. in Kelly's book) needs $V$ to be closed, because it asks for isomorphisms

$$C(v * c, c') \cong v \multimap C(c, c')$$

where $C(c, c')$ is the hom-object, and $\multimap$ is the closed structure of $V$.

But $V$ doesn't need to be closed if you instead ask for natural bijections

$$V(v', C(v * c, c')) \cong V(v' \otimes v, C(c, c'))$$

(Symmetry isn't needed at all for this.)

Uh, I never noticed this split of conventions, I always used the second

Matteo Capucci (he/him) (Apr 22 2022 at 13:02):

Dylan McDermott said:

Tim Campion said:

Hm... is there some kind of adjunction between $V$-categories and $V$-actegories?

The way I prefer to think about this is that all of these things embed into Wood's "large V-categories" (which modulo size issues are categories enriched over $[V^{\mathrm{op}}, \mathbf{Set}]$ with Day convolution). Wood shows that there is a fully faithful 2-functor from $V$-categories to $[V^{\mathrm{op}}, \mathbf{Set}]$-categories, and the image of this is the $[V^{\mathrm{op}}, \mathbf{Set}]$-categories $C$ for which $C(c, c') : V^{\mathrm{op}} \to \mathbf{Set}$ is representable for all $c, c'$ (the representations are the hom-objects). There is a similar fully faithful 2-functor from the 2-category of left $\mathrm{V}$-actegories, strong functors and strong natural transformations, and the image is those $C$ for which $C(c, {-})v : C_0 \to \mathbf{Set}$ is representable. ($C_0$ is the underlying ordinary category.)

When $C(c, {-})v$ and $C(c, c')$ are representable (so there is both an enrichment and an actegory), the representations form an adjunction, and this is one way of proving the theorem about closed actegories and tensored $V$-categories.

Now this is interesting :thinking:

Tim Campion (Apr 22 2022 at 15:24):

Alexander Campbell said:

Tim Campion I give a treatment of this comparison from the "enrichment as structure" POV in Section 4 of my paper on Skew-enriched categories. The idea is that you can see V-categories and V-actegories as special kinds of skew V-proactegories. (N.B. To capture V-categories you really do need to consider skew V-proactegories!)

How have I not read this paper!!? (well -- I know how -- I've seemingly become allergic to reading lately! And, yes, "lately" encompasses back at least 5 years :) This looks great... Just starting, but already at Example 2.3 I'm intrigued -- I was more expecting a V-category with object set A in the usual sense to be a skew-enrichment of the _codiscrete_ category on A, rather than the discrete one! (basically just because the way it works when you think in terms of polyads...)

Tim Campion (Apr 22 2022 at 15:30):

I'm puzzled right now also because there seems to be a tension between @Dylan McDermott 's comment and @Alexander Campbell 's comment -- in Dylan's setup we're thinking about what I think would be reasonably called a "pro-enriched category", whereas Alexander's parenthetical seems to indicate that such a setting will not be sufficient.

Tim Campion (Apr 22 2022 at 15:40):

Side question: I think there's a functor $(-)-Cat : SymMulti-n-Cat \to SymMulti-(n+1)-Cat$ for each $n$, and in particular there's an endofunctor $(-)-Cat : SymMulti-\omega-Cat \to SymMulti-\omega-Cat$. (The point being that these are constructions you can iterate.) To what extent can you iterate "taking enriched categories" if you want to allow non-symmetric-multicategories? Or if you want to encompass enrichment in a bicategory? There's probably at least something to say using fc-multicategory / virtual double category type machinery -- though ideally it would be nice to have a story you can tell without that much machinery, and then find that it lifts to that more sophisticated setting.

Nathanael Arkor (Apr 22 2022 at 15:58):

Or if you want to encompass enrichment in a bicategory?

Categories enriched in a bicategory $\mathcal W$ form a bicategory $\mathcal W\text{-}\mathbf{Cat}$, so I think it's reasonable to drop "symmetric monoidal" and move the base of enrichment up one dimension instead, in which case you get endofunctors at every dimension (considering bicategories enriched in tricategories, and so on).

Nathanael Arkor (Apr 22 2022 at 16:01):

This then essentially becomes the topic (for $n = 2$) of Garner–Shulman's Enriched categories as a free cocompletion.

Tim Campion (Apr 22 2022 at 16:05):

Yeah, I should probably slow down and think sometimes when I write, because as soon as I wrote the above, I did realize I should be thinking about Garner and Shulman... But it's funny, because the enrichment functor they consider is idempotent, since it's a free cocompletion with respect to some absolute colimits. Whereas I'm imagining something which is interesting to iterate. The difference must lie in the choice of morphisms -- if you take $V$-profunctors as your morphisms of $V$-categories, the process is idempotent, whereas if you take $V$-functors as morphisms, it isn't. And this is reflected in the fact that the version of the construction they give with a whole proarrow equipment is not idempotent -- it's still a free-cocompletion, but with respect to weights which are no longer absolute.

Tim Campion (Apr 22 2022 at 16:09):

Another thing to think about is that they work with locally cocomplete bicategories -- I don't know what their story looks like when you relax the local cocompleteness. My motivation in putting things in terms of symmetric multicategories rather than symmetric monoidal categories above was to try to isolate something that might make sense without such cocompleteness hypotheses... I think I'm trying to generalize in too many directions at once!

Tim Campion (Apr 22 2022 at 16:11):

It's probably not the best idea to try to understand categories enriched in a small lax skew category -oid all in one step :upside_down:

Nathanael Arkor (Apr 22 2022 at 16:13):

Yes, I've also wondered what can be said when you drop local cocompleteness.

Mike Shulman (Apr 22 2022 at 16:42):

Dylan McDermott said:

The usual definition of "tensored" (e.g. in Kelly's book) needs $V$ to be closed, because it asks for isomorphisms

$$C(v * c, c') \cong v \multimap C(c, c')$$

where $C(c, c')$ is the hom-object, and $\multimap$ is the closed structure of $V$.

But $V$ doesn't need to be closed if you instead ask for natural bijections

$$V(v', C(v * c, c')) \cong V(v' \otimes v, C(c, c'))$$

Of course, the latter natural bijection says precisely that $C(v * c, c')$ has the universal property of an internal-hom $v \multimap C(c, c')$. So if you interpret the former isomorphism as the statement that the RHS exists and the LHS is isomorphic to it, they are equivalent.

Mike Shulman (Apr 22 2022 at 16:44):

If you drop local cocompleteness, then you can't compose V-profunctors any more, so you don't get a bicategory of them. At that level of generality, I think the construction is naturally viewed as operating on [[virtual equipments]].

Tim Campion (Apr 22 2022 at 16:49):

Mike Shulman said:

If you drop local cocompleteness, then you can't compose V-profunctors any more, so you don't get a bicategory of them. At that level of generality, I think the construction is naturally viewed as operating on [[virtual equipments]].

Right... maybe it's simpler first to think about the "monads" construction rather than the "enriched categories" construction. At the level of virtual equipments, you and Geoff gave a universal property for this construction -- iirc it's right adjoint to the inclusion of normal virtual double categories into all virtual double categories. Which is now confusing because I was expecting a left adjoint somehow...

Nathanael Arkor (Apr 22 2022 at 17:07):

Presumably it's possible in theory to derive the universal property for $\mathcal V \mapsto \mathcal V\text{-}\mathbf{Cat}$ using the universal properties of the matrix construction and the monads construction. Though this may not be at all practical.

Mike Shulman (Apr 22 2022 at 17:47):

The way I think of it is that the forgetful functor $U : \rm NormalVDC \to VDC$ has a right adjoint $\rm Mod$, and therefore the composite ${\rm Mod} \circ U$ is a monad on $\rm NormalVDC$. This monad, call it $T$, is then also essentially the same as the free $T$-algebra functor, which is of course left adjoint to the forgetful functor $T \rm Alg \to NormalVDC$.

Mike Shulman (Apr 22 2022 at 17:49):

In other words, $\rm Mod$ is a right adjoint when defined on $\rm VDC$, but a left adjoint when defined on $\rm NormalVDC$. This makes sense because in order to have a unit $V\to {\rm Mod}(V)$ you need $V$ to have units.