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Stream: theory: category theory

Topic: laxer limits

Matteo Capucci (he/him) (Mar 24 2022 at 11:19):

Fix a bicategory $\mathbb X$. The lax limit of a diagram $D: \mathbb I \to \mathbb X$ in $\mathbb X$ is defined to be the pseudorepresenting objects of the functor

$$\mathcal L(X) = [I, Cat]_l(1, \mathbb X(X, D-))$$

meaning that

$$\mathbb X({\lim}_{lax} D, X) \simeq \mathcal L(X).$$

We could weaken this notion even more by replacing the latter equivalence with an adjunction. Is this a known definition?
In concrete terms, this would mean to weaken the universal property of $\lim_{lax} D$ as to hold 'up to 2-cells'. For instance, a laxer product of two objects $X$ and $Y$ would look like a normal product, except that given $f:Z \to X$ and $g:Y \to Z$, the universal morphism $\langle f,g \rangle_{lax}$ does not factor $f$ and $g$ through the projections strictly, but only up to a 2-cell:
image.png

Matteo Capucci (he/him) (Mar 24 2022 at 11:21):

In other words, $(\langle f,g \rangle_{lax}, l_X)$ is the (left?) Kan extension of $f$ along $\pi_X$, and likewise $(\langle f,g \rangle_{lax}, l_Y)$ is the Kan extension of $g$ along $l_Y$.

Reid Barton (Mar 24 2022 at 11:30):

There is a very similar question somewhere on MathOverflow.
I admit I don't quite understand the motivation--such a "universal property" would only determine the limit "up to adjunction", or something--is that the kind of thing you want?

Matteo Capucci (he/him) (Mar 24 2022 at 12:10):

I don't want it, but I've stumbled upon 'laxer coproducts' in the wild, and I thought they were lax colimits until I realized laxness was somewhere else
Later I will post here the example I have

Matteo Capucci (he/him) (Mar 24 2022 at 13:10):

Reid Barton said:

There is a very similar question somewhere on MathOverflow.

By the way, do you have some reference to the MO question?

Reid Barton (Mar 24 2022 at 13:37):

https://mathoverflow.net/questions/386414/very-lax-2-dimensional-co-limits

Reid Barton (Mar 24 2022 at 13:51):

In the "In concrete terms" section, you don't mean that the "universal" morphism is the (essentially) unique morphism together with 2-cells as indicated, but only the initial such morphism, right?

Reid Barton (Mar 24 2022 at 13:56):

So, for example, the "laxer limit" of the empty diagram in Cat could be any category with an initial object.

Jacques Carette (Mar 24 2022 at 14:04):

That seems kind of cool -- shades of Hilbert's $\epsilon$ operator, no? A neat way to pick out the categories with initial objects.

Matteo Capucci (he/him) (Mar 24 2022 at 16:19):

Reid Barton said:

https://mathoverflow.net/questions/386414/very-lax-2-dimensional-co-limits

Thanks!

Matteo Capucci (he/him) (Mar 24 2022 at 16:20):

Reid Barton said:

In the "In concrete terms" section, you don't mean that the "universal" morphism is the (essentially) unique morphism together with 2-cells as indicated, but only the initial such morphism, right?

Exactly

Matteo Capucci (he/him) (Mar 24 2022 at 16:20):

Reid Barton said:

So, for example, the "laxer limit" of the empty diagram in Cat could be any category with an initial object.

Very cool!

Mike Shulman (Mar 24 2022 at 16:38):

We probably shouldn't refer to it with "the", then...