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Stream: learning: questions

Topic: ✔ Why are Pointwise Kan Extensions called that?

Chris Grossack (they/them) (Jul 24 2023 at 18:17):

Say $\mathcal{A} \overset{j}{\to} \mathcal{B}$ is an inclusion of $\mathcal{A}$ as a subcategory of $\mathcal{B}$. Then given a functor $f : \mathcal{A} \to \mathcal{X}$ we might want to extend it to a functor defined on $\mathcal{B}$. The universal way to do this is called the Left Kan Extension of $f$ along $j$, and this extension tends to exist (for instance, whenever $\mathcal{A}$ is "small enough" and $\mathcal{X}$ is cocomplete).

Chris Grossack (they/them) (Jul 24 2023 at 18:20):

Now, a (left) kan extension is called a Pointwise Kan Extension if it's one that we can really compute, for instance by a formula involving coends. There's a definition (given in Loregian's Coend Calculus) that a kan extension is pointwise exactly when it plays nice with comma categories in the sense shown in this diagram (taken from the book):

Screenshot-2023-07-24-at-11.20.31-AM.png

Chris Grossack (they/them) (Jul 24 2023 at 18:21):

My question, then, is this:

Why is this called a "pointwise" kan extension? I might guess that it's because we can compute it "pointwise" with the coend formula... But this condition on comma categories is somewhat mysterious to me, and it seems like a weird choice to take as the definition

Mike Shulman (Jul 24 2023 at 18:43):

Yes, it's because of the coend formula: the value of the Kan extension functor at each object (i.e. at each point) can be computed with its own universal property, rather than just the entire functor having a universal property.

Mike Shulman (Jul 24 2023 at 18:45):

The connection of the coend formula to the comma category definition is that if you take $X=1$ then the comma category definition tells you that the "evaluation" of the extension at a "point" $k: 1 \to C$ has its own universal property, which expressed as a colimit (= Kan extension to 1) over that comma category, which turns out to be equivalent to the usual coend definition.

Mike Shulman (Jul 24 2023 at 18:45):

The general version is the usual sort of thing where you replace elements by generalized elements.

Mike Shulman (Jul 24 2023 at 18:47):

It's worth noting that the comma-category definition is equivalent to the coend-based definition in "set-like" 2-categories such as $\rm Cat$, or internal categories in some category with finite limits, but is not equivalent to it in other 2-categories such as categories enriched over some monoidal category. In the latter case, the coend definition is the right one, not the comma category definition. But there is an abstract version of the coend definition that makes sense in any proarrow equipment or Yoneda structure.

Chris Grossack (they/them) (Jul 24 2023 at 19:10):

Mike Shulman said:

The connection of the coend formula to the comma category definition is that if you take $X=1$ then the comma category definition tells you that the "evaluation" of the extension at a "point" $k: 1 \to C$ has its own universal property, which expressed as a colimit (= Kan extension to 1) over that comma category, which turns out to be equivalent to the usual coend definition.

Oooooooooooooh. That makes perfect sense, thanks ^_^

Notification Bot (Jul 24 2023 at 19:10):

Chris Grossack (they/them) has marked this topic as resolved.