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Stream: learning: questions

Topic: Preservation of domains, codomains and identities

Gabriel Goren Roig (Nov 28 2023 at 16:37):

Hi! There's a passage in Emily Riehl's CT in Context that I find confusing. It's in Chapter 3, when discussing completeness of Cat. Here it is:

Screenshot_20231128_132515.png

She says that if the limit of a diagram X of small categories existed, not only the set of objects (resp. morphisms) of that category should be the limit of the diagram of sets of objects (resp. morphisms), but also that "domains, codomains and identities are also preserved" because we can think of them as the natural transformations dom:Cat(2,C) -> Cat(1,C), cod:Cat(2,C) -> Cat(1,C) and id:Cat(1,C) -> Cat(2,C) arising from the 3 functors between ordinal categories 1 and 2.

I understand that last statement (after "because"), but I don't understand what "preservation of domains, codomains and identities" means exactly nor why it is implied by that.

Intuitively, I guess what is meant is that if I take a morphism f in the limit category and look at its domain, that domain should be "the limit" of the domains of the morphisms "whose limit" is f. But I don't know how to make this precise.

Kevin Arlin (Nov 28 2023 at 17:40):

A diagram of categories involves, among other things, a diagram of arrows $s_i: M_i\to O_i$ where $M_i,O_i$ are the morphism sets in the $i$'th category of the diagram and $s_i$ is the domain-assigning function. That is, this is a diagram in the category whose objects are arrows in the category of sets, and thus it's perfectly sensible to take its limit to see that "the domain function of the limit category is the limit of the domain functions." Another viewpoint is that $s_i$ gives a natural transformation between the diagrams of morphisms and of objects; then the point is that taking limits is a functor, that is, sends natural transformations to functions. So that's another way to see the domain function in the limit as the limit of the domain functions in the diagram.

Gabriel Goren Roig (Nov 28 2023 at 20:57):

Thanks @Kevin Arlin ! Your second viewpoint seems to be what Riehl was pointing at. I'm trying to formulate precisely the claim that "domains are preserved" using the action on morphisms (natural transformations) of the functor $\text{lim}_J : [J, \text{Set}] \to \text{Set}$.

I think the claim is that given a diagram $X: J \to \text{Cat}$, the function $\text{dom}_{\text{lim}X}: \text{Cat}(2, \text{lim}X) \to \text{Cat}(1, \text{lim}X)$ (the component at $\text{lim}X$ of natural transformation $\text{mor}$) is equal to the composite $\text{Cat}(2, \text{lim}X) \simeq \text{lim}(\text{mor}X) \xrightarrow[]{\text{lim}(\text{dom}X)} \text{lim}(\text{ob}X) \simeq \text{Cat}(1, \text{lim}X)$. Does this seem right to you?