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Stream: theory: category theory

Topic: How is this "initial object" called?

Bruno Gavranović (Jun 09 2023 at 20:03):

I have a construction that resembles a lax/weak/local initial object and, because I'm not sure if a good reference for these things exists, I'm not sure how to disambiguate between them and determine what this is. Let me be precise.

I have a bicategory $\mathcal{B}$ and an object $I : \mathcal{B}$ such that a few different things hold:

• For every object $X$, the hom-category $\mathcal{B}(I, X)$ has an initial object $i_X: I \to X$ (i.e. for every other other $g : I \to X$ there is a unique 2-cell $i_x \Rightarrow g$);
• For every $f : X \to Y$ in $\mathcal{B}$ there is a unique 2-cell $i_X ; f \Rightarrow i_Y$

The second condition makes me think that this is a lax initial object, as the commuting condition isn't satisfied strictly.
On the other hand, I feel like there's a missing "...such that" condition to this structure that I haven't quite figured out yet.
Likewise, I'm not sure whether every "lax initial object" (whose definition I'm still uncertain of) will locally be a strict initial object, as I have here.

Bruno Gavranović (Jun 11 2023 at 18:11):

While talking to @Igor Bakovic I realised my problem specification is incorrect in a manner where it makes it clear what this construction is.
Note that the second bullet point has a map going into the initial object $i_y$ which is a little bit suspicious. As it turns out, I made a mistake and my setting has a map going out of it. This then means that this bullet point is redudant.

Then, looking at the remaining bullet point, the condition of $\mathcal{B}(I, X)$ having an initial object means that there exists a left adjoint to the terminal functor $\mathcal{B}(I, X) \to \mathbf{1}$.
This generalises the setting of an initial object in a category (where we have an isomorphism $\mathcal{C}(I, X) \cong 1$, with $\mathcal{C}$ being any category) to one in a bicategory (where now we have an adjunction).

When we generalise isomorphisms to adjunctions this is often called a quasi-colimit . So my bicategory really has a quasi-initial object.

Some thoughts I had while realising this:

• I don't understand why this is called a quasi limit and not a lax limit. The latter name seems more apt as we only have 2-cells going in one direction.
• It looks like a quasi-initial object $I$ in $\mathcal{B}$ implies the existence of a strict colimit in every hom-category $\mathcal{B}(I, X)$. I never heard about this fact before, but it sounds important to know.
• More surprisingly, even, it seems like this holds the other way too. Having a strict colimit in every such hom-category $\mathcal{B}(I, X)$ seems to imply that $\mathcal{B}$ has a quasi-initial object, as postcomposition with $f : X \to Y$ lands us in $\mathcal{B}(I, Y)$ for which already have proof that it contains a map from the initial object to any other.

Mike Shulman (Jun 11 2023 at 18:15):

I believe the terminology "quasi" is due to Gray, who was writing before the standard meaning of "lax" as "up to a noninvertible transformation" had been adopted. I don't think these things have been studied further sufficiently for a consistent modern terminology to be adopted. But the terminology [[lax limit]] is not available because that already means something different.

Mike Shulman (Jun 11 2023 at 18:16):

(In particular, a "lax initial object" is the same as an ordinary initial object.)

Igor Bakovic (Jun 12 2023 at 17:02):

Jay gave an elementary description of the kind of conical limits of morphisms of bicategories which arise from local adjunctions in

C.B. Jay, Local adjunctions, Journal of Pure and Applied Algebra 53, (1988) 227-238.

In Example 2.7. he defines a conical limit of a morphism $F \colon \mathcal{D} \to \mathcal{A}$ of bicategories by means of a local right adjoint to the diagonal (strict) homomorphism

$\Delta \colon \mathcal{A} \to Bicat^{op}(\mathcal{D},\mathcal{A})$

where $Bicat^{op}(\mathcal{D},\mathcal{A})$ is the bicategory whose objects are morphisms of bicategories, 1-cells are oplax transformations and 2-cells are modifications. Then a limit for $F$ is a lax cone $\alpha \colon \Delta (X) \to F$ such that, for each lax cone $\beta \colon \Delta (Y) \to F$ there is a pair $(f, \sigma)$ where $f \colon Y \to X$ is a l-cell in $\mathcal{A}$ and $\sigma \colon \alpha \circ \Delta (f) \to \beta$ is a modification, which is universal among such pairs in the sense that for any other such pair $(f', \sigma')$ there exists a unique 2-cell $\chi \colon f' \to f$ such that $\sigma'=\sigma(\alpha \circ \Delta (\chi))$.

In the special case when $\mathcal{D}$ is $\emptyset$ a conical limit $T$ of the empty diagram $\emptyset \to \mathcal{A}$ is an object with a property of a local terminal object - an object such that for each object $A$ there is a terminal object in the hom-category $\mathcal{A} (A, T)$.

Therefore, what you have is a dual situation - a local initial object.

I could say much more about the role of local initial and terminal objects in local and lax adjunctions but perhaps I should stop here for now.