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I am reading some handwritten notes, and scribbled next to a restriction of scalars functor, are the words "a pullback".

I don't understand why this might be the case.

In particular, consider a field $k$ with a map $\varphi:k\rightarrow A$ for $A$ an associative unital $k$-algebra. Then we consider the induced functor $\varphi_*:A-mod\rightarrow k-mod$ restriction of scalars.

Is $\varphi_*$ the pullback of some diagram in the category of functors from $A$-mod to $k$-mod?

I know this a fairly strange question, but I just cant see what the scribbler had in mind.

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I'm not sure I follow here -- don't you mean that $\varphi_* : A-\text{mod} \rightarrow k-\text{mod}$? – Daniel McLaury Nov 13 '10 at 22:36
I most definitely do, I will edit. Thanks. – BBischof Nov 13 '10 at 22:36
and should the question have also been changed? – Arturo Magidin Nov 13 '10 at 23:29
sigh, yes it should have – BBischof Nov 13 '10 at 23:39
up vote 3 down vote accepted

The sense in which "pullback" is being used here is the same sense it's being used in this Wikipedia article; that is, it's being used to refer to the process of "pulling back" a morphism $A \to \text{End}(R)$ (an $A$-module) along the morphism $k \to A$ to get a morphism $k \to \text{End}(R)$ (a $k$-module) by precomposition. This is different from the categorical pullback. I don't know why they're named the same.

Edit: Okay, so now I do know why they're named the same. To be brief, I think the historical motivation is that the pullback of vector bundles can be defined in both ways; see, for example, this blog post by Akhil Mathew.

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