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Stream: theory: category theory

Topic: explicit factorizations in 2Cat

Daniel Teixeira (Mar 27 2024 at 17:30):

In the canonical model category Cat we can get a very explicit description of the factorization systems $(\text{Cof}\cap W,\text{Fib}$ and $\text{Cof},\text{Fib}\cap W$ (in e.g. Rezk's note on the model structure). Is there anything like this for the Lack model structure in 2Cat?

Kevin Carlson (aka Arlin) (Mar 27 2024 at 18:01):

Lack does most of this in the original paper. Fibrations are equiv-fibrations, equivalences are bi-equivalences, cofibrant objects are exactly those with free underlying category. Cofibrations don't get quite as an explicit characterization as isocofibrations do in the case of Cat; they're maps whose underlying functor lifts against full+surjective-on-objects functors, though, which is at least better than what you get straight from the construction of the model structure.

Daniel Teixeira (Mar 27 2024 at 19:40):

I mean given a 2-functor $F$ between 2-categories, an explicit description of the factorization as e.g. a trivial cofibration followed by a fibration

Daniel Teixeira (Mar 27 2024 at 19:41):

Lack's papers don't do that because, as with most model categories, his argument factors through some stronger result

Daniel Teixeira (Mar 27 2024 at 19:45):

But for instance, given any functor $F:C\to D$ between categories, an explicit factorization is given as follows. Let $\hat C$ be the categories whose objects are tuples $(c,d,f:Fc\to d$, where $f$ is an isomorphism, the hom-set between $(c,d,f)$ and $(c',d',f')$ is $\hom_C(c,c')$. Define $G:C\to \hat C$ by sending $c$ to $(c,Fc,\text{id}_{Fc}$, and $H:\hat C\to D$ by sending $(c,d,f)$ to $d$. Then you can check that $F = H\circ G$, that $G$ is a trivial cofibration (equivalence surjective on objects)< and that $H$ is an isofibration.

Daniel Teixeira (Mar 27 2024 at 19:48):

I was wondering if we have something similar for 2Cat, but calculations with my colleague are suggesting that the obvious generalization doesn't work.

(The obvious generalization starts with a 2-functor $F:C\to D$ and $\hat C$ is defined as the 2-category which has $(c,d, (f,g,\varepsilon,\eta))$ as objects, where $(f,g,\varepsilon,\eta)$ is adjoint equivalence data between $Fc$ and $d$)