Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: lax limit: data for 2-cells

Matteo Capucci (he/him) (Aug 05 2021 at 13:53):

From "Co/end calculus"
The definition consumes a 2-functor $F$ but only uses its action on 0- and 1-dimensional data.

Is this correct?
If I had to write down the definition of 'laxly lax' limit (???), i.e. a 3-dimensional limit-like object which specifies also 2-dimensional structure for $\mathrm{llim} F$ , would I use the action of $F$ on 2-cells?

Matteo Capucci (he/him) (Aug 05 2021 at 13:57):

I imagine something like this: in addition to the 1-dimensional and 2-dimensional data, one would also require that for every $\alpha : f \Rightarrow g$ in $\mathcal A$ , there is an 3-morphism $\Pi_\alpha : \pi_f \to Fg \circ F\alpha$

Reid Barton (Aug 05 2021 at 14:27):

This definition isn't complete; there should be some coherence conditions on the $\pi_f$

Matteo Capucci (he/him) (Aug 05 2021 at 15:25):

There are, but I'md
Matteo Capucci (he/him) said:

From "Co/end calculus"

If you're talking about this, there is another piece of the def in the next page but it's about the UP and I'm interested in the data for now

Matteo Capucci (he/him) (Aug 05 2021 at 15:25):

Matteo Capucci (he/him) said:

I imagine something like this: in addition to the 1-dimensional and 2-dimensional data, one would also require that for every $\alpha : f \Rightarrow g$ in $\mathcal A$ , there is an 3-morphism $\Pi_\alpha : \pi_f \to Fg \circ F\alpha$

If you're talking about this, what shall they be?

Reid Barton (Aug 06 2021 at 10:59):

Matteo Capucci (he/him) said:

There are, but I'md
Matteo Capucci (he/him) said:

From "Co/end calculus"

If you're talking about this, there is another piece of the def in the next page but it's about the UP and I'm interested in the data for now

Yes, there is a missing condition on the data. As the nlab puts it here:

the assignment [ $f \mapsto \pi_f$ ] behaves sensibly with respect to identities and composition (see the references for details).

Matteo Capucci (he/him) (Aug 07 2021 at 08:54):

Mmh I see

Matteo Capucci (he/him) (Aug 07 2021 at 08:54):

I'm not completely following the reason why

Matteo Capucci (he/him) (Aug 07 2021 at 08:56):

My guess is: cones with tip $X$ over $F$ are given by $Nat(\Delta_X, F)$ , so going one dimension up, and laxly so, we get that lax cones with tip $X$ over $F$ are given by $LaxNat(\Delta_X, F)$ ? Therefore $(p, \pi)$ has to satisfy the axioms of a lax natural transformation?