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Stream: theory: category theory

Topic: Smallest dense subcategories

Emily (Aug 06 2021 at 20:06):

A subcategory $\mathcal{A}$ of a category $\mathcal{C}$ is dense if the inclusion functor $\iota\colon\mathcal{A}\hookrightarrow\mathcal{C}$ is so, satisfying any of the following equivalent conditions:

image.png

Now, define the smallest dense full subcategory of $\mathcal{C}$ to be, up to equivalence¹, the full subcategory $\mathcal{A}$ of $\mathcal{C}$ satisfying the following pair of conditions:

1. The category $\mathcal{A}$ is dense in $\mathcal{C}$.
2. If $\mathcal{A}'$ is another dense full subcategory of $\mathcal{C}$, then either $\mathcal{A}'$ contains $\mathcal{A}$ or it is equivalent¹ to $\mathcal{A}$.

(¹Maybe replacing equivalence by isomorphism might be interesting, too (?))

Question. What are some examples of these? In particular, does $\mathcal{C}=\mathsf{CRing}$ admit a smallest dense full subcategory (with dense being understood in the enriched category sense in this case)?

Fawzi Hreiki (Aug 06 2021 at 20:42):

I'll wager a guess that $\mathcal{C}$ is the smallest dense subcategory of $\text{PSh}(\mathcal{C})$.

Fawzi Hreiki (Aug 06 2021 at 20:52):

Regarding the $\text{CRing}$ case, the best bet is probably the full subcategory on the object $\mathbb{Z}[x, y]$. This works because the theory of rings has a presentation with max arity 2.

John Baez (Aug 07 2021 at 00:53):

It's sot of jumping the gun to say the smallest dense full subcategory, as you did, before you know it exists.

John Baez (Aug 07 2021 at 00:57):

I guess I'm sounding like a broken record, because I've said this before, but it makes my skin crawl when people use "the" in situations when it's not known that there's a contractible $\infty$-groupoid of things with the given property. (In the case of sets this means "the" element exists and is unique; for higher categories the uniqueness gets suitably softened up.)

John Baez (Aug 07 2021 at 00:58):

E.g. saying "the first aliens that landed on Earth and helped humanity".

John Baez (Aug 07 2021 at 01:02):

Or defining "the smallest dense subcategory" when you don't know one exists.

John Baez (Aug 07 2021 at 01:02):

In both cases it feels like rhetorical overreach: trying to persuade us into accepting the existence of something just by acting like it exists. :upside_down:

Martti Karvonen (Aug 07 2021 at 01:25):

For instance, I'd wager that the full subcategory on $\mathbb{Z}[x,y,z]$ is also dense in $\text{CRing}$, so I'm not sure "the smallest dense subcategory" exists . By the way, what's the enrichment for $\text{CRing}$ that you have in mind?

Emily (Aug 07 2021 at 02:59):

@John Baez That's a very good point! I was under the impression that this thing always existed since the identity functor is dense, but of course condition 2) is still a problem. Here's a patched definition (these may still not exist, but the following is nevertheless a better definition than the previous one):

Definition. A minimal dense full subcategory of $\mathcal{C}$ is, if it exists, a dense full subcategory $\mathcal{A}$ of $\mathcal{C}$ that is also a minimal element in the poset of dense full subcategories of $\mathcal{C}$, ordered by inclusion of categories.

Emily (Aug 07 2021 at 02:59):

The previous definition read the same, but with "least" instead of "minimal"

Emily (Aug 07 2021 at 02:59):

@Martti Karvonen and @Fawzi Hreiki Thanks for the examples! I'm thinking of the enrichment of $\mathsf{CRing}$ where $\mathbf{Hom}_{\mathsf{CRing}}(R,S)$ is the ring of ring maps from $R$ to $S$, equipped with pointwise addition and multiplication.

John Baez (Aug 07 2021 at 03:25):

Théo said:

Definition. A minimal dense full subcategory of $\mathcal{C}$ is, if it exists, a dense full subcategory $\mathcal{A}$ of $\mathcal{C}$ that is also a minimal element in the poset of dense full subcategories of $\mathcal{C}$, ordered by inclusion of categories.

That's better. Until we prove otherwise, minimal dense full subcategories may not exist and they may not be unique.

An even better approach, probably, is to admit that we have not only inclusions of subcategories, but also natural isomorphisms between these, so if we really want a poset we should look at equivalence classes of subcategories where two subcategories $i: A \to C$, $i': A' \to C$ are equivalent if there's an equivalence $j: A \to A'$ such that $i' j$ is naturally isomorphic to $i$. These form a poset in a natural way.

Martti Karvonen (Aug 07 2021 at 03:59):

Théo said:

Martti Karvonen and Fawzi Hreiki Thanks for the examples! I'm thinking of the enrichment of $\mathsf{CRing}$ where $\mathbf{Hom}_{\mathsf{CRing}}(R,S)$ is the ring of ring maps from $R$ to $S$, equipped with pointwise addition and multiplication.

I assumed that rings are unital and homomorphisms should preserve the unit. If so, adding two ring homomorphisms pointwise is not in general a ring homomorphism. I don't think that multiplying homomorphisms pointwise results in a homomorphism either (this time the end result might not preserve sums).

Matteo Capucci (he/him) (Aug 07 2021 at 08:58):

John Baez said:

It's sot of jumping the gun to say the smallest dense full subcategory, as you did, before you know it exists.

OT: what sort of linguistic device shall I use then? Just 'a'?

Morgan Rogers (he/him) (Aug 07 2021 at 09:08):

John Baez said:

An even better approach, probably, is to admit that we have not only inclusions of subcategories, but also natural isomorphisms between these, so if we really want a poset we should look at equivalence classes of subcategories where two subcategories $i: A \to C$, $i': A' \to C$ are equivalent if there's an equivalence $j: A \to A'$ such that $i' j$ is naturally isomorphic to $i$. These form a poset in a natural way.

I'm pretty sure that's what "subcategory" normally means: an equivalence class of inclusion (i.e. full and faithful) functors, the same way "subset" means an equivalence class of injective functions. Although I suppose one could look at isomorphism equivalence classes rather than "equivalence equivalence classes" :joy:

Morgan Rogers (he/him) (Aug 07 2021 at 09:13):

Fawzi Hreiki said:

I'll wager a guess that $\mathcal{C}$ is the smallest dense subcategory of $\text{PSh}(\mathcal{C})$.

$\mathcal{C}$ can only be recovered from $\text{PSh}(\mathcal{C})$ up to idempotent completion (and even that only up to equivalence), so any subcategory whose idempotent completion is the same as that of $\mathcal{C}$ provides a dense subcategory. In particular, I'm sure it wouldn't be hard to come up with an idempotent-complete category with no smallest generating subcategory.

Morgan Rogers (he/him) (Aug 07 2021 at 09:38):

Here you go: consider the category with an object $[n]$ for each $[n] \in \mathbb{N}$. Each $[n]$ has endomorphisms $e^{[n]}_0,\dotsc,e^{[n]}_n$ such that $e^{[n]}_ie^{[n]}_j = e^{[n]}_je^{[n]}_i = e^{[n]}_{\max\{i,j\}}$ (so $e_0$ is the identity morphism). Then we add basic morphisms $d^{[n]}_+: [n] \to [n+1]$ and $d^{[n]}_-: [n] \to [n-1]$ such that $d^{[n+1]}_- d^{[n]}_+ = e^{[n]}_0$ and $d^{[n]}_+ d^{[n+1]}_- = e^{[n+1]}_1$, and similarly $d^{[n+1]}_+ d^{[n]}_+ d^{[n+1]}_- d^{[n+2]}_- = e^{[n+2]}_2$ etc. The idea is that this is the union of the finite categories obtained as the idempotent completions of the finite total orders, viewed as idempotent monoids.

Any infinite full subcategory of this one is dense, but there is no minimal such; I'll leave you the fun task of checking whether this is also the case in the category of presheaves over this category.

Fawzi Hreiki (Aug 07 2021 at 11:23):

Perhaps it’s best then to consider subcategories which are (at least) closed under colimits

Morgan Rogers (he/him) (Aug 07 2021 at 11:28):

Huh? Isn't a dense subcategory which is closed under colimits automatically the whole category?

Fawzi Hreiki (Aug 07 2021 at 12:13):

Sorry - typo. I meant idempotent splitting

Fawzi Hreiki (Aug 07 2021 at 12:19):

Either way, there has to be a more appropriate notion of size rather than just subcategory inclusion. For example the full subcategories $\{\mathbb{Z}[x, y]\}, \{\mathbb{Z}[x, y, z]\} \hookrightarrow \text{Ring}$ are both dense while neither contains the other. Yet it still feels like the first is obviously 'smaller' than the second.

Martti Karvonen (Aug 07 2021 at 14:08):

There's a natural monomorphism between the inclusions, going from the 'smaller' to the larger one

Fawzi Hreiki (Aug 07 2021 at 14:20):

You're right actually. Well are there any examples of incomparable minimal dense subcategories?

Morgan Rogers (he/him) (Aug 07 2021 at 14:32):

Any object in one dense subcategory must be a colimit of objects and morphisms in the other, so there will always be some kind of comparison.

Martti Karvonen (Aug 07 2021 at 16:32):

Morgan Rogers (he/him) said:

Any object in one dense subcategory must be a colimit of objects and morphisms in the other, so there will always be some kind of comparison.

How does this work in practice? Say we take full subcategories on $\{\mathbb{Z}[x, y, z]\}$ and on $\{\mathbb{Z}[x, y,],R\}$ where $R$ is some ring that doesn't admit monomorphisms to/from $\mathbb{Z}[x, y, z]$? I guess there will be some natural transformation between the inclusions, but if it's not monic or some such, it's unclear if it makes sense to read the comparison as $\leq$ in a way that let's us talk about "smallest/minimal" dense subcategories.

Morgan Rogers (he/him) (Aug 07 2021 at 16:37):

Indeed, I would be surprised if the concept of minimal dense subcategory is something which exists in general. I was just pointing out that you can't avoid there being morphisms in the ambient category between the objects of such categories... although I now realise that including the initial object in one subcategory and not the other, if the initial object is strict, would make even that observation untrue.

Morgan Rogers (he/him) (Aug 07 2021 at 16:39):

(you would need an example with a few more objects to show that the morphisms need not assemble into any kind of natural transformation)

John Baez (Aug 08 2021 at 02:43):

Matteo Capucci (he/him) said:

John Baez said:

It's sot of jumping the gun to say the smallest dense full subcategory, as you did, before you know it exists.

OT: what sort of linguistic device shall I use then? Just 'a'?

Yes. I think "a minimal dense subcategory" sounds good because we've got a poset here and the question of whether a minimal element exists is up for grabs, along with whether it's unique.

John Baez (Aug 08 2021 at 02:43):

Somehow "smallest" suggests uniqueness-if-it-exists to me, while "minimal" does not.

John Baez (Aug 08 2021 at 02:45):

For example, it sounds weird to say "Thumbelina is a smallest horse in the world", even if there are several horses of equally small size.

John Baez (Aug 08 2021 at 02:46):

Similarly, "a smallest subcategory" sounds a bit odd; I think mathematicians say "minimal" to avoid this.