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Stream: learning: questions

Topic: Exponentials in (1,2)-presheaves

Harrison Grodin (Dec 18 2025 at 21:57):

I'm considering the category "(1,2)-presheaves", functors $B^\text{op} \to \mathbf{Poset}$, where homomorphisms between two functors are [[lax natural transformations]].

In particular, I'm studying the case where $B$ is the walking arrow $\{0 \leftarrow 1\}$, ie the "lax arrow category"(?) of $\mathbf{Poset}$: a homomorphism from $X_0 \xrightarrow{\chi} X_1$ to $Y_0 \xrightarrow{\psi} Y_1$ is a pair of maps $f_i : X_i \to Y_i$ (for $i = 0,1$) such that $f_1 \circ \chi \le \psi \circ f_0$.

As far as I can tell, this category has pointwise limits and colimits, but the laxness is getting in my way when checking if this category has exponentials. (Typically, exponentials in presheaf categories can be found using the Yoneda lemma, but the analogous fact is only an adjunction, not an equivalence.) Does anyone know if this category has exponentials?

Kevin Carlson (Dec 18 2025 at 22:19):

It doesn't have pointwise limits! If it did, evaluation at $0$ would probably be representable, but it's not. The problem is that $\chi$ doesn't have to map the equalizer of one pair to the equalizer of the others, under laxity.

Harrison Grodin (Dec 18 2025 at 22:22):

Agh, rookie mistake - I checked products and incorrectly analogized to pullbacks. Thanks Kevin!!!

Notification Bot (Dec 18 2025 at 22:22):

Harrison Grodin has marked this topic as resolved.

Kevin Carlson (Dec 18 2025 at 22:22):

We got stuck on whether a closely analogous category has equalizers at all sometime last year here: #learning: questions > Is the category of pseudo-squares locally presentable? @ 💬

Kevin Carlson (Dec 18 2025 at 22:22):

I still don't know the answer.

Harrison Grodin (Dec 18 2025 at 22:24):

Interesting...

Harrison Grodin (Dec 18 2025 at 22:26):

The broader reason I was curious about this is that I have a use for this "lax arrow category" (building on Section 4.5 here), and I was wondering if this category might have more structure than I initially realized - although apparently, it has less! :sweat_smile:

Harrison Grodin (Dec 18 2025 at 22:29):

(It is the Kleisli category for a monad on the "real" arrow category $\mathbf{Poset}^{\to}$, with strict natural transformations - called $\mathcal{S}$ in the linked paper, Remark 4.15. So, it inherits the colimits automatically, and also products, but evidently not much else?)

Kevin Carlson (Dec 19 2025 at 00:00):

Well, categories of algebras and lax morphisms for a 2-monad are decent in some sense. The pseudo ones, especially, have PIE limits and are generally friendly in a way well-understood by Bourke as mentioned in the thread I linked above. I don't remember what the lax map categories have exactly but I believe it's known.

Notification Bot (Dec 19 2025 at 00:00):

Kevin Carlson has marked this topic as unresolved.

John Baez (Dec 19 2025 at 09:55):

In math nothing is ever "resolved": we keep understanding things better and better. I would like a version of Zulip where pushing the "resolved" button releases a burst of laughter.