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Stream: theory: category theory

Topic: Right adjoint to inclusion functor

Nathaniel Virgo (Apr 10 2023 at 07:13):

In my work I have the following situation: category $\mathcal{D}$ is a (non-full) subcategory of category $\mathcal{C}$. In fact the objects of $\mathcal{D}$ consist of those objects of $\mathcal{C}$ that have a certain property, and the morphisms of $\mathcal{D}$ are those morphisms of $\mathcal{C}$ that have a certain property.

In my case the inclusion $\mathcal{D}\hookrightarrow\mathcal{C}$ has a right adjoint. I'd like to look at other examples where this happens, to get a feel for what they have in common. So I'm wondering if there are some other nice examples where an inclusion functor has a right adjoint.

nlab has a list of 'cofree' functors at [[free functor]], which are right adjoints to forgetful functors, but in those examples the forgetful functor is forgetting structure and not just properties, so they're not quite what I'm looking for.

One example I know of is the case of representable Markov categories, where the inclusion functor $\mathcal{C}_\text{det}\hookrightarrow\mathcal{C}$ has a right adjoint given by the distribution functor $P$. But still, I'm wondering if there are other nice examples (not necessarily probability related), and/or theorems that are relevant to this situation.

Amar Hadzihasanovic (Apr 10 2023 at 07:33):

When $\mathcal{D}$ is a full subcategory, this is called a [[coreflective subcategory]]. The nLab has a list of examples. I guess you can also dualise the examples at [[reflective subcategory]].
Not sure about non-full examples.

Mike Shulman (Apr 10 2023 at 08:07):

If your "inclusion" functor is not full, then by definition (at least according to the catechism of [[stuff, structure, property]]) it is forgetting structure and not just properties. Even if it happens accidentally to be injective on objects, that property will be lost if you replace its domain by an equivalent category, so it is not a "categorical property".

Nathaniel Virgo (Apr 10 2023 at 08:40):

Mike Shulman said:

If your "inclusion" functor is not full, then by definition (at least according to the catechism of [[stuff, structure, property]]) it is forgetting structure and not just properties. Even if it happens accidentally to be injective on objects, that property will be lost if you replace its domain by an equivalent category, so it is not a "categorical property".

I'm puzzling about the right way to think about this in regards to my example. I don't want to explain my example as it's a bit complicated, but a similar example is the inclusion functor $\mathcal{C}_\text{det}\to\mathcal{C}$ of the deterministic subcategory of a Markov category into its underlying Markov category. Intuitively, it feels like it's forgetting a property, although it's a property of the morphisms rather than the objects (namely that they are deterministic). But maybe we can say it's actually forgetting the comonoid structure on the objects?

Reid Barton (Apr 10 2023 at 08:51):

I think this is a case of "property-like structure".

Amar Hadzihasanovic (Apr 10 2023 at 09:02):

I think that yes, you can see $\mathcal{C}_\mathrm{det}$ as, equivalently, the category of “comonoids in $\mathcal{C}$ which are equal to the chosen comonoids”, i.e. comonoid-preserving functors from the walking comonoid qua Markov category to $\mathcal{C}$, together with comonoid homomorphism.
As Reid said the structure of a “comonoid equal to the chosen comonoid” is, somewhat by definition, property-like.

Mike Shulman (Apr 10 2023 at 09:04):

I don't know what a Markov category or its deterministic subcategory is, but I can say something about some examples that I do understand. Consider for instance the category $\rm Met$ of metric spaces and continuous maps, and its non-full subcategory $\rm Met_{\le}$ of nonexpansive maps. In this case I would say the structure that's being forgotten is that of a metric. But, I hear you ask, don't the objects of $\rm Met$ already have metrics? Well, yes, because we defined them that way, but that metric is actually superfluous data: $\rm Met$ is equivalent to the category $\rm Top_{met}$ of metrizable topological spaces and continuous maps, and this is a "more honest" definition of that category because the morphisms preserve all the structure that's present on the objects. Now the functor $\rm Met_{\le} \to Top_{met}$ is no longer injective on objects, but it is still faithful and we can see it more clearly as forgetting the structure of a metric.

Mike Shulman (Apr 10 2023 at 09:06):

Conversely, any functor that forgets structure can be viewed as a non-full subcategory in a simliar way. For instance, consider the forgetful functor $\rm Grp \to Set$. The category $\rm Set$ is equivalent to the category whose objects are groups and whose morphisms are arbitrary functions between their underlying sets. This category, in turn, contains $\rm Grp$ as a non-full subcategory. But clearly the usual version of this functor $\rm Grp \to Set$ that is not injective on objects is more "honest" at showing how it forgets group structures.

Mike Shulman (Apr 10 2023 at 09:08):

(Note that neither of these examples is property-like.)

Amar Hadzihasanovic (Apr 10 2023 at 09:12):

(One thing to keep in mind in Nathaniel's example is that a Markov category is itself a “category with extra structure” so the notions of functor, subcategory, etc. have to be adapted to structure-preserving ones. In particular the notion of “deterministic subcategory” is not invariant under isomorphism of underlying categories.)

Reid Barton (Apr 10 2023 at 09:14):

I also don't know what a Markov category is but I assumed the examples were things like convex spaces.

Reid Barton (Apr 10 2023 at 09:16):

There's a difference in behavior between convex spaces and say vector spaces--a (free) vector space has a lot of different bases, so if we want to talk about maps between vector spaces that are induced by a map of basis vectors, then obviously we have to put extra structure on the vector spaces to do that. But the generators of a free convex space (i.e. a simplex) are determined by the space up to isomorphism, they're the vertices of the simplex. So if "deterministic map" means one that preserves "being a vertex" then it's not as obvious that there is a choice of extra structure involved.

Reid Barton (Apr 10 2023 at 09:17):

Same thing happens e.g. for free commutative monoids.

Nathaniel Virgo (Apr 10 2023 at 10:37):

Thank you all, there's a lot to think about here.

Here's an example that's similar to the Markov category example but simpler if you don't know about Markov categories. Let $T$ be a monad on a category $\mathcal{C}$ and consider the functor $\mathcal{C}\to \mathrm{Kl}(T)$ that sends $f\colon X\to Y$ to $f;\eta\colon X\to TY$, where $\eta$ is the monad unit. This can be seen as a kind of inclusion if we regard $\mathcal{C}$ as a subcategory of $\mathrm{Kl}(T)$, namely the one composed of all those morphisms that have the form $f;\eta$ for some $f\in\mathcal{C}_1$. ("Representable Markov categories" are a specific case where this inclusion functor has a right adjoint, which is the underlying functor of $T$.) The functor seems like it's forgetting a property of morphisms (namely that they are of the form $f;\eta$), but presumably there is also a way to regard it as forgetting structure on the objects instead.

Reid Barton (Apr 10 2023 at 11:35):

The category $\mathrm{Kl}(T)$ is equivalent to the category of free $T$-algebras, as a full subcategory of all $T$-algebras.
In general, nonisomorphic objects of $\mathcal{C}$ may become isomorphic in $\mathrm{Kl}(T)$ under this "inclusion", so that's a strong indication that an object of $\mathcal{C}$ has "extra structure". Specifically, that structure is a choice of generators for the corresponding free $T$-algebra.

Reid Barton (Apr 10 2023 at 11:40):

Sorry a better thing to say than "nonisomorphic objects become isomorphic" would have been "many isomorphisms in $\mathrm{Kl}(T)$ do not belong to the subcategory $\mathcal{C}$".

Nathaniel Virgo (Apr 10 2023 at 11:53):

Ah, cool, now I understand what you meant about simplices

Reid Barton (Apr 10 2023 at 11:58):

Right, so in the convex spaces example (which I guess is a typical sort of example for you, though I don't actually know), this:

Reid Barton said:

Sorry a better thing to say than "nonisomorphic objects become isomorphic" would have been "many isomorphisms in $\mathrm{Kl}(T)$ do not belong to the subcategory $\mathcal{C}$".

doesn't happen, so there it is more reasonable to think that the objects of both categories contain the same information somehow, we just consider two different kinds of maps in the two categories.

Mike Shulman (Apr 10 2023 at 14:51):

Note that for a general monad $T$, the functor $C \to {\rm Kl}(T)$ is not even faithful, so there is no sense in which it can be called an "inclusion".

Mike Shulman (Apr 10 2023 at 14:52):

Amar Hadzihasanovic said:

(One thing to keep in mind in Nathaniel's example is that a Markov category is itself a “category with extra structure” so the notions of functor, subcategory, etc. have to be adapted to structure-preserving ones. In particular the notion of “deterministic subcategory” is not invariant under isomorphism of underlying categories.)

I presume that what you meant is that it's not invariant under equivalence of underlying categories. I can't imagine that it isn't invariant under isomorphism.

Mike Shulman (Apr 10 2023 at 14:57):

From a fully category-theoretic point of view, one can argue that a notion that isn't invariant under equivalence of categories shouldn't be called a "category with extra structure". A category with extra structure is something like a monoidal category, equipped with "category-theoretic" structure that is necessarily so invariant. Which isn't to say that there's anything wrong with noninvariant notions, it's just that they should really be regarded as category-like structures, with (as you say) their own notions of functor and so on -- including "substructure" -- rather than categories with structure.

Amar Hadzihasanovic (Apr 10 2023 at 15:17):

Perhaps I was imprecise in my phrasing, but I simply meant that "there are non-isomorphic Markov categories (with non-isomorphic deterministic subcategories) with isomorphic underlying categories", similarly to how there are non-isomorphic groups with isomorphic underlying sets.

I did not mean that the structure of Markov category cannot be transported along equivalences -- I think it can, that is, it is genuinely "extra structure on a category" and not a "category-like structure".

Mike Shulman (Apr 10 2023 at 15:39):

Generally when we say that something is "invariant under isomorphism" we mean that it can be transported across isomorphisms. Thus, group structure is invariant under isomorphism because if $G$ is a group and its underlying set $UG$ is isomorphic to some other set $X$, there is a unique induced compatible group structure on $X$. Thus, a structure that's "not invariant under isomorphism" is one that doesn't have this property.

Mike Shulman (Apr 10 2023 at 15:41):

Similarly for "invariant under equivalence". Ordinary categorical structure, like a monoidal structure on a category, is invariant under equivalence in this sense.

Mike Shulman (Apr 10 2023 at 15:46):

I've looked up the definition of [[Markov category]], and indeed I don't think it is invariant under equivalence in this way. I believe it's a special case of a [[supply in a monoidal category]], and in general supplies are not equivalence-invariant. I think the counterexamples should specialize to Markov categories: let $C$ be a semicartesian monoidal category containing an object $X$ with two different commutative comonoid structures, let $C'$ be a category equivalent to $C$ with two copies of $X$, and make $C'$ a Markov category in which the two copies of $X$ have different comonoid structures (hence the isomorphism between them is not deterministic). Then there shouldn't be any Markov structure on $C$ making the equivalence $C'\simeq C$ a Markov functor.

John Baez (Apr 10 2023 at 16:11):

@Mike Shulman wrote:

If your "inclusion" functor is not full, then by definition (at least according to the catechism of [[stuff, structure, property]]) it is forgetting structure and not just properties.

@Nathaniel Virgo

Intuitively, it feels like it's forgetting a property, although it's a property of the morphisms rather than the objects...

When Mike said "forgetting structure" he meant "forgetting structure on objects" - that's the default meaning.

In general, "forgetting a property of morphisms" is the same as "forgetting a structure on objects".

So, there's no conflict here.

For example, the forgetful functor from semigroups to sets is essentially surjective - every set can be made into a semigroup - but it is not full: there are usually fewer homomorphisms from one semigroup to another than functions between their underlying sets. So, we could say this functor "forgets properties on morphisms", but it's also true, and more common, to say it "forgets structure" - that is, forgets the semigroup structure on objects.

John Baez (Apr 10 2023 at 16:16):

Anyone curious about nuances of this "property, structure, stuff" business check out the paper that Mike and I wrote about it - Section 2.4, in particular.

John Baez (Apr 10 2023 at 16:18):

I think it's really helpful to use these terms in a precise way, or at least be able to.