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Stream: theory: category theory

Topic: Nice terminology for 2-dimensional mono/epimorphisms

Emily (Apr 01 2024 at 15:32):

I'm writing some things involving the 2-dimensional analogues of monomorphisms and epimorphisms in a bicategory, but there doesn't seem to be an established terminology for all the notions I want to consider, so I'm having a bit of a hard time coming up with a nice set of names for these.

First, let me recall 8 basic analogues of mono/epi's in bicategories. Recall that a morphism $f\colon A\to B$ in a category $\mathcal{C}$ is a monomorphism if

$$f_{*} \colon \mathrm{Hom}_{\mathcal{C}}(X,A) \to \mathrm{Hom}_{\mathcal{C}}(X,B)$$

is injective and it is an epimorphism if

$$f^{*} \colon \mathrm{Hom}_{\mathcal{C}}(B,X) \to \mathrm{Hom}_{\mathcal{C}}(A,X)$$

is injective. So, when moving from categories to bicategories, we replace injective with either 1) faithful, 2) full, 3) fully faithful, or 4) ask that the underlying morphism of sets is injective. This leads to a total of 8 notions of monos/epis in bicategories. I've been calling these "faithful/full/fully faithful/strict mono/epimorphisms", so that e.g. a morphism $f\colon A\to B$ in a bicategory $\mathcal{C}$ for which

$$f^{*} \colon \mathsf{Hom}_{\mathcal{C}}(B,X) \to \mathsf{Hom}_{\mathcal{C}}(A,X)$$

is a faithful functor would be termed a "faithful epimorphism".

However, this terminology is not completely nice in that e.g. if we talk about "faithful epimorphisms" in the 2-category of categories, it is then not clear whether we mean the notion as above or an epimorphism of categories that is also a faithful functor.

Meanwhile, a different choice would be to call these co/representably faithful/full/fully faithful as is done e.g. in the nLab here, though that also runs into problems as there are other clashing notions of such morphisms (see Section 3 there). In addition, the mention of "mono/epimorphism" in the name is gone, so the terminology itself doesn't signalise that these are 2-dimensional analogues of mono/epis.

Meanwhile again, yet another different choice is to call a "fully faithful epimorphism" a "lax epimorphism", as is done e.g. here, here, or here. This runs into the problem that: 1) there's no term for "full epimorphism" 2) what would be an "oplax epimorphism"?

Meanwhile yet again, there are a lot of other notions of epi/monos in 2-categories, based on working with isomorphisms instead of morphisms ("pseudomonic", as done here), using lifting properties with respect to lax epimorphisms as in Definition 3.4 here, or adapting the definitions of e.g. regular/split/etc. mono/epi's to 2-categories.

All in all, what would be a nice and comprehensive set of terminology for the many, many notions of mono/epi's in bicategories?

Nathanael Arkor (Apr 01 2024 at 16:36):

Emily said:

Meanwhile, a different choice would be to call these co/representably faithful/full/fully faithful as is done e.g. in the nLab here, though that also runs into problems as there are other clashing notions of such morphisms (see Section 3 there). In addition, the mention of "mono/epimorphism" in the name is gone, so the terminology itself doesn't signalise that these are 2-dimensional analogues of mono/epis.

Regarding this terminology in particular, I don't think "(co)representably fully faithful" is ambiguous at all: rather, it is the term "fully faithful morphism" (without the prefix "representably") that is ambiguous/misleading.

Nathanael Arkor (Apr 01 2024 at 16:37):

Personally, I think the "representably X" naming convention is quite nice, because it's clear and unambiguous. Though I do agree it means one loses the conceptual connection to epis/monos.

Nathanael Arkor (Apr 01 2024 at 16:38):

In Proper factorization systems in 2-categories, Dupont and Vitale suggest a naming convention for at least some of these variations (i.e. $(i, j)$-properness, see page 2 and 3).

Emily (Apr 01 2024 at 17:27):

Nathanael Arkor said:

Personally, I think the "representably X" naming convention is quite nice, because it's clear and unambiguous. Though I do agree it means one loses the conceptual connection to epis/monos.

Incidentally, come to think about it, "co/representable monomorphisms/epimorphisms" as in morphisms such that $f_*$ or $f^*$ are monomorphisms/epimorphisms might give yet another possibly interesting notion. Have you ever seen these notions pop up anywhere, Nathanael?

Mike Shulman (Apr 01 2024 at 17:45):

I agree that there's no problem with the "representably" terminology. Another systematic naming scheme is

• representably fully faithful = 1-monomorphism
• representably faithful = 2-monomorphism
• corepresentably fully faithful = 1-epimorphism
• corepresentably faithful = 2-epimorphism

This doesn't include "representably full" morphisms, but those are extremely rare; I don't think I've ever seen anyone use them for anything. (Your other class of morphisms that are "representably injective on objects" is not really even a sensible definition in a bicategory, as it is not invariant under equivalence of bicategories.)

Mike Shulman (Apr 01 2024 at 17:46):

"Lax" should be avoided at all costs, since it has a standard meaning of "up to a noninvertible 2-cell".

Mike Shulman (Apr 01 2024 at 17:46):

It's also not uncommon to abbreviate "representably fully faithful" by just "fully faithful" and "corepresentably fully faithful" by "co-fully-faithful".

Emily (Apr 01 2024 at 17:59):

Mike Shulman said:

I agree that there's no problem with the "representably" terminology. Another systematic naming scheme is

• representably fully faithful = 1-monomorphism
• representably faithful = 2-monomorphism
• corepresentably fully faithful = 1-epimorphism
• corepresentably faithful = 2-epimorphism

This doesn't include "representably full" morphisms, but those are extremely rare; I don't think I've ever seen anyone use them for anything. (Your other class of morphisms that are "representably injective on objects" is not really even a sensible definition in a bicategory, as it is not invariant under equivalence of bicategories.)

I think 2-monomorphisms would probably clash with monomorphisms in $\mathsf{Hom}_{\mathcal{C}}(A,B)$ =/

In any case, I'm trying to write up a comprehensive set of notes on these, so I actually do want to cover representably full morphisms as well

Jonas Frey (Apr 01 2024 at 18:01):

There are also pseudo-monomorphisms!

Emily (Apr 01 2024 at 18:02):

Jonas Frey said:

There are also [[pseudo-monomorphisms]]!

(This is not related to the current discussion, but I was reading your note here about this these days and found it really helpful/useful. Thanks for writing it!)

Jonas Frey (Apr 01 2024 at 18:04):

Jonas Frey said:

There are also pseudo-monomorphisms!

These are precisely the maps f with "trivial kernel", ie where the sqare with two identities at the source and two times f towards the sink is a pseudopullback

Emily (Apr 01 2024 at 18:05):

Jonas Frey said:

There are also pseudo-monomorphisms!

I guess it might also make sense to consider full/faithful/fully faithful functors like

$$\mathsf{Iso}_{\mathcal{C}}(B,X)\to\mathsf{Iso}_{\mathcal{C}}(A,X)$$

The number of potentially interesting notions of monos/epis in bicategories just gets worse the longer you think about it, it seems :sweat_smile:

Jonas Frey (Apr 01 2024 at 18:07):

Emily said:

(This is not related to the current discussion, but I was reading your note here about this these days and found it really helpful/useful. Thanks for writing it!)

Thanks! Yes, that was inspired by Baez-Shulman's "Lectures on $n$-categories and cohomology". There I saw for the first time the idea of (co)representably defining $n$-monos and $n$-epis in higher categories.

Kevin Carlson (aka Arlin) (Apr 01 2024 at 18:12):

Regarding the representably full morphisms, in the 1-categorical case the representably subjective morphisms are often a special case of strong epis, namely when there’s an initial object maps out of which are mono. Strong epis in general are almost like maps which remain representably surjective when lifted into coslices, but that monicity condition for maps from initials becomes very rare there upon coslicing…not a complete thought, just something about how representably full maps might show up.

Emily (Apr 01 2024 at 19:15):

Incidentally, I think having the map

$$f^{*} \colon \mathsf{Iso}_{\mathcal{C}}(B,X) \to \mathsf{Iso}_{\mathcal{C}}(A,X)$$

be full gives yet another possibly interesting notion to the list: if $\phi\circ f\cong\psi\circ f$ via a 2-iso $\alpha$, then $\phi\cong\psi$ via a 2-iso $\beta$, and these isomorphisms are compatible (i.e. $\alpha=\beta\star\mathrm{id}_{f}$)

Mike Shulman (Apr 01 2024 at 22:23):

I think a "representably surjective" map in a 1-category is the same as a split epi.

Mike Shulman (Apr 01 2024 at 22:24):

@Emily, I'll be surprised if you find anything interesting to say about representably full morphisms.

Mike Shulman (Apr 01 2024 at 22:24):

Emily said:

I think 2-monomorphisms would probably clash with monomorphisms in $\mathsf{Hom}_{\mathcal{C}}(A,B)$ =/

What does that notation mean?

Kevin Carlson (aka Arlin) (Apr 01 2024 at 22:54):

Oops, strong epis are left orthogonal to monos :innocent:

Mike Shulman (Apr 02 2024 at 01:19):

In general, I think it's kind of an accident that the corepresentably injective morphisms in Set and other especially well-behaved 1-categories (like topoi and abelian categories) are a notion of "surjection". In higher categories, and in many other 1-categories (like Ring), a better notion of "surjection" is a strong epi, which as you say is left orthogonal to the monos.

Mike Shulman (Apr 02 2024 at 01:22):

In Cat and other similar 2-categories, there are two similar factorization systems to (strong epi, mono), namely (essentially surjective, fully faithful) and (essentially surjective and full, faithful). This is one reason why the just-full morphisms are not usually very useful or interesting: they're an odd mix of "surjectivity" and "injectivity" type conditions. The "injectivity" type conditions (faithful and fully-faithful) are best characterized representably, while the "surjectivity" type conditions (essentially-surjective and essentially-surjective-and-full) are best characterized by orthogonality to those. An in-between condition like fullness (or the opposite mix of "essentially surjective and faithful") can then be characterized by mixing these two factorization systems into a [[ternary factorization system]].

John Onstead (Apr 02 2024 at 03:08):

In general, what is the relationship between k-surjective morphisms and k-epimorphisms?

Mike Shulman (Apr 02 2024 at 04:33):

With the terminology as at those pages, a morphism between $\infty$-groupoids is a k-epimorphism just when it is j-essentially-surjective for $0\le j\le k-1$. However, I don't really like that usage of "k-epimorphism" because it invites confusion with morphisms that are co-representably [[n-monomorphisms]].

Nathanael Arkor (Apr 02 2024 at 07:50):

Emily said:

Incidentally, come to think about it, "co/representable monomorphisms/epimorphisms" as in morphisms such that $f_*$ or $f^*$ are monomorphisms/epimorphisms might give yet another possibly interesting notion. Have you ever seen these notions pop up anywhere, Nathanael?

So, asking for certain functors to be monomorphisms/epimorphisms in Cat? I suspect considering epimorphisms will not give you a particularly nice concept, since I don't believe there's an explicit characterisation of epic functors. On the other hand, the monomorphisms are the functors that are injective on objects and morphisms, so it doesn't seem unreasonable to consider these. However, at that point, it may be sensible to work up to equivalence and just consider the representably fully faithful morphisms instead.

Emily (Apr 02 2024 at 15:18):

Mike Shulman said:

Emily, I'll be surprised if you find anything interesting to say about representably full morphisms.

I probably won't! Mostly I'm trying to do this for completeness. The main thing I want to try is to look at the representably full morphisms in some 2-categories and see if there are nice characterisations of them, like $\mathsf{Rel}$, $\mathsf{Span}$, $\mathsf{Cats}_{\mathsf{2}}$, $\mathsf{Mod}$, $\mathsf{Prof}$, etc.

For $\mathsf{Rel}$ I've found the one here and for $\mathsf{Cats}$ have asked about a nicer one here, as the currently-known characterisation is really messy. I haven't tried figuring out the other ones yet.

Emily (Apr 02 2024 at 15:19):

Mike Shulman said:

Emily said:

I think 2-monomorphisms would probably clash with monomorphisms in $\mathsf{Hom}_{\mathcal{C}}(A,B)$ =/

What does that notation mean?

I think "2-monomorphism" might clash with the notion of a 2-morphism in a bicategory which happens to be a monomorphism in its Hom-category

Emily (Apr 02 2024 at 15:44):

Emily said:

Incidentally, I think having the map

$$f^{*} \colon \mathsf{Iso}_{\mathcal{C}}(B,X) \to \mathsf{Iso}_{\mathcal{C}}(A,X)$$

be full gives yet another possibly interesting notion to the list: if $\phi\circ f\cong\psi\circ f$ via a 2-iso $\alpha$, then $\phi\cong\psi$ via a 2-iso $\beta$, and these isomorphisms are compatible (i.e. $\alpha=\beta\star\mathrm{id}_{f}$)

Incidentally, but how bad would the name "iso-representably full morphism" be for those? Is there a nicer choice?

Morgan Rogers (he/him) (Apr 02 2024 at 16:00):

The "iso" should definitely be attached to the "full" and not to the "representably"!

Nathanael Arkor (Apr 02 2024 at 16:01):

Representably [[conservative]] morphisms are similar, if not the same.

Emily (Apr 02 2024 at 16:06):

Morgan Rogers (he/him) said:

The "iso" should definitely be attached to the "full" and not to the "representably"!

My first instinct was that it would be better to attach it to "representably X", since "representably" means the "X" condition happen for the functor on the $\mathsf{Hom}_{\mathcal{C}}$ categories, so then "iso-representably X" would mean the "X" condition happens for the functor on the $\mathsf{Iso}_{\mathcal{C}}$ categories, where here X = full, but could be anything.

Do you agree/disagree with this?

Morgan Rogers (he/him) (Apr 02 2024 at 16:26):

I think iso-full is a sensible abbreviation of "full on isomorphisms", which is what you're going for.

Emily (Apr 02 2024 at 16:33):

Morgan Rogers (he/him) said:

I think iso-full is a sensible abbreviation of "full on isomorphisms", which is what you're going for.

Ooh right! That makes a lot of sense!

Mike Shulman (Apr 02 2024 at 17:13):

You can also use a prefix "core-" for something relating to isomorphisms.

Emily (Apr 02 2024 at 21:46):

Mike Shulman said:

You can also use a prefix "core-" for something relating to isomorphisms.

This made me realise that besides

$$f^{*} \colon \mathsf{Iso}_{\mathcal{C}}(B,X) \to \mathsf{Iso}_{\mathcal{C}}(A,X)$$

one also has

$$f^{*} \colon \mathsf{Core}(\mathsf{Hom}_{\mathcal{C}}(B,X)) \to \mathsf{Core}(\mathsf{Hom}_{\mathcal{C}}(A,X)).$$

It's asking the second $f^*$ here to be full that leads to the (more) interesting "monomorphism-like" notion where $\phi\circ f\cong\psi\circ f$ implies $\phi\cong\psi$ with a condition of the form $\beta=\alpha\star\mathrm{id}_f$ relating the two 2-isomorphisms.

Emily (Apr 02 2024 at 21:48):

So I guess the least ugly-sounding name for these might be "corepresentably full on cores"