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Stream: learning: questions

Topic: Grothendieck Construction

Fabrizio Genovese (Apr 12 2020 at 22:22):

I just had this idea. I am pretty sure someone has already come up with something similar, so I'd appreciate any reference if you have some!
For what I was able to find around the Grothendieck construction is basically defined for functors to $\textbf{Cat}$ and functors to $\textbf{Set}$, in which case we call it "category of elements. I need something in the middle, that is, an analogous construction for functors $F: \mathcal{C} \to \mathcal{D}$. If $\mathcal{D}$ has a terminal object $1$, then we can proceed as follows: Define $\int_\mathcal{C} F$ as the category having:
Objects: Pairs $(C, 1 \xrightarrow{x} FC)$
Morphisms: A morphism $(C, 1 \xrightarrow{x} FC) \to (D, 1 \xrightarrow{y} FD)$ is a morphism $f: C \to D$ such that $x;Ff = y$.
In the case $\mathcal{D} = \textbf{Set}$ or $\mathcal{D} = \textbf{Cat}$, we should get back the original definition. It's nothing special really, but unless I missed something it works and it's quite cute. Is there something like this in the literature already (again I'm pretty sure there is, I just don't know where)? :slight_smile:

Nathanael Arkor (Apr 12 2020 at 22:27):

Isn't this the comma category $1_{\mathcal D} \downarrow F$? (Where $1_{\mathcal D} : 1 \to \mathcal D$ is the functor sending $* \mapsto 1_{\mathcal D}$.)

Fabrizio Genovese (Apr 12 2020 at 22:30):

What is the constantly terminal functor?

Fabrizio Genovese (Apr 12 2020 at 22:33):

Oh, that one

Fabrizio Genovese (Apr 12 2020 at 22:34):

Shoudn't it be the opposite? The comma category $1_\mathcal{D} \downarrow F$?

Fabrizio Genovese (Apr 12 2020 at 22:36):

(In any case, now that you let me notice this, I think so. But I never thought of the Grothendieck construction as something related to comma categories)

Fabrizio Genovese (Apr 12 2020 at 22:37):

(Maybe this observation is trivial for people working in pure CT, but I didn't know this and it's very nice!)

Fabrizio Genovese (Apr 12 2020 at 22:41):

Ok, the relationship with the category of elements seems to be worked out here: https://ncatlab.org/nlab/show/functors+and+comma+categories