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Stream: theory: category theory

Topic: lax coslices and local presentability

Tom Hirschowitz (Jul 02 2024 at 14:02):

I know nothing about locally presentable 2-categories, but it would be a strong incentive for me if the following analogy worked.

• The categories $𝐒𝐞𝐭$ and $1/𝐒𝐞𝐭$ are locally presentable, and the codomain functor $1/𝐒𝐞𝐭 → 𝐒𝐞𝐭$ is accessibly monadic (hence in particular it creates limits).
• The category of elements of a continuous, accessible functor $F\colon 𝐂 → 𝐒𝐞𝐭$ with $𝐂$ locally presentable, is its pullback with the codomain functor $1/𝐒𝐞𝐭 → 𝐒𝐞𝐭$.
• In this situation, $F$ is representable, and its category of elements is isomorphic to the coslice under the representing object.
• Since locally presentable categories are closed under coslices, this readily entails that all categories of elements of continuous, accessible functors $F\colon 𝐂 → 𝐒𝐞𝐭$ for a locally presentable $𝐂$ are again locally presentable. Furthermore, their projections to $𝐂$ are accessibly monadic.

Now, for a 2-category $𝐂$ and $c ∈ 𝐂$, let $c ⫽ 𝐂$ denote the ‘oplax coslice’, i.e.,

• objects are pairs of an object $d$ and a morphism $c → d$,

• morphisms $(d,f) → (d',f')$ are pairs $(g,α)$ as in the following diagram
  oplaxslice.png

  

• 2-cells $(g,α) → (h,β)$ are 2-cells $γ\colon g → h$ in $𝐂$ making the obvious pasting commute.

Do the above points hold upon replacing

• $𝐒𝐞𝐭$ with $𝐂𝐚𝐭$,
• $1/𝐒𝐞𝐭$ with $1⫽𝐂𝐚𝐭$,
• category of elements with Grothendieck construction,
• local presentability with its 2-categorical variant?

Tom Hirschowitz (Jul 02 2024 at 14:46):

Mmmm... quite unlikely to work, since $1⫽𝐂𝐚𝐭$ does not have an initial object, nor a bi-initial object, which I guess is necessary for being locally presentable, right?

Ivan Di Liberti (Jul 03 2024 at 10:45):

Tom Hirschowitz said:

Mmmm... quite unlikely to work, since $1⫽𝐂𝐚𝐭$ does not have an initial object, nor a bi-initial object, which I guess is necessary for being locally presentable, right?

I never thought about it, but if it's true that it does not have a binitial object, it will never be bipresentable with respect to the definition that is Osmonds and mine.

Tom Hirschowitz (Jul 03 2024 at 12:09):

Ok, thanks for confirming!

Kevin Carlson (Jul 03 2024 at 12:32):

The identity of $1$ is still an initial object in some interesting lax sense, though--the category $A=1//\mathbf{Cat}(\mathrm{id}_1,p)$ is neither isomorphic to nor equivalent to $1,$ but it does have a terminal object, or in more generalizable terms, $A$ admits a final functor from $1.$ I wonder what formalism makes this an appropriate kind of "weak initial object".

Nathanael Arkor (Jul 03 2024 at 13:01):

One subtlety about the 2-dimensional setting is that there are multiple notions of local presentablity one might consider. For a finitely complete 2-category $\mathcal K$, the models of $\mathcal K$ should be 2-functors preserving the finite limits. However, one has a choice of which kinds of natural transformations to take between the 2-functors. Strict or pseudo are obvious choices, but one could also take lax natural transformations, in which case one obtains a notion of local presentability which is very different from the 1-dimensional setting and which, in particular, one does not necessarily have (co)completeness.

Kevin Carlson (Jul 03 2024 at 17:37):

The lax slice is precisely the 2-category of models of the theory of a pointed object in the sense of (op, I think)lax transformations, right? But then, without cocompleteness, to what extent is this a reasonable notion of local presentability at all? Is it somehow a correct Gray-notion of local presentability?

Nathanael Arkor (Jul 03 2024 at 19:19):

But then, without cocompleteness, to what extent is this a reasonable notion of local presentability at all?

If your definition of "locally presentable category" is "the category of models for a finite limit theory", then it's a one possible generalisation of this definition to the 2-dimensional setting, in which there is now a choice of what to take as the 2-cells. In other words, it might be reasonable to have four different notions of local presentability in two dimensions, depending on the kind of 2-cells one considers. Taking strict natural transformations will give the most well-behaved notion. Pseudo natural transformations will still be fairly well-behaved and cover more natural examples. Op/lax natural transformations will give 2-categories that are less well-behaved, but may still be of interest in practice (e.g. I imagine they should have some of the op/lax notions of limit/colimit described in Štěpán's Colax adjunctions and lax-idempotent pseudomonads).

Nathanael Arkor (Jul 03 2024 at 19:20):

I would imagine the laxer notion of locally presentable 2-category to satisfy some "oplax adjoint functor theorem", for instance.

Nathanael Arkor (Jul 03 2024 at 19:21):

(But this is all highly speculative, as I don't believe anyone has studied these notions yet. Even 2-categories of algebras and lax morphisms are much less studied than their pseudo counterparts.)

Kevin Carlson (Jul 03 2024 at 22:14):

Thanks for the thoughts. I think an oplax adjoint functor theorem would be quite satisfying.

Tom Hirschowitz (Jul 04 2024 at 07:02):

Thanks, Nathanael and Kevin, this is quite instructive. The motivation for the question lies in 2-dimensional initial-algebra semantics, for which 2-categories of models need to have some sort of initial object. So this probably disqualifies the op/lax variants.