Re: Direct image functors

[George Janelidze <janelg@telkomsa.net>]

Dear Peter and Steve,

If anyone protests against the direct image functor being a right adjoint, I
would recall the following simple well-known story:

1. If f : X-->Y is a map of sets, then the inverse image (=pullback) functor

f* : Sets/Y --> Sets/X has both left and right adjoint. Let me call them Lf
and Rf, respectively.

2. If we replace Sets with an arbitrary category C with pullbacks, then f*
and Lf are still there, but Rf disappears, unless C is locally cartesian
closed. In particular, there is no Rf (in general) when C = Top is the
category of topological spaces.

3. But if I am thinking towards topos theory, I might prefer to consider not

f* : Top/Y --> Top/X,

but

f* : Shv(Y) --> Shv(X), where Shv(?), the category of sheaves (of sets) over
"?", is equivalent to the full subcategory of Top/? with objects all local
homeomorphisms with codomain "?", and, under this equivalence, the 'new' f*
is the restriction of the old one. And then we have f* and Rf but not Lf (in
general).

Therefore, I might prefer to have name "direct image functor" for the right
adjoint! Peter, might this be what you thought of as "geometric aspects"?
(Surely, a geometer, considering, say, a manifold X, would be more
interested in Shv(X) than in Top/X.)

With apologies for trivialities-
George

--------------------------------------------------
From: "Steve Vickers" <s.j.vickers@cs.bham.ac.uk>
Sent: Tuesday, November 8, 2022 4:10 PM
To: <ptj@maths.cam.ac.uk>
Cc: <categories@mta.ca>
Subject: categories: Re:  Direct image functors

>
> Dear Peter,
>
> I agree the term isn???t likely to change (to ???sections functor??? or anything
> else) at this stage. I was partly trying to find out how widely the issue
> was recognised, and partly trying to sharpen my discussion of it.
>
>> I don't think anyone is likely to be deceived into thinking that it's a
>> direct
>> image in the set-theoretic sense.
>
> I???m not so sure. I???ve seen how when people start looking more closely at
> the points of a topos, and the part they play in topological analogies,
> that there is a risk of confusion. I have known a student, learning  about
> the action of a geometric morphism on points, who wondered if it???s somehow
> closely related to the direct image functor.
>
> By the way, I looked at the Elephant to see what you said there, and I saw
> ???we shall see later that, in a sense, f_* ???embodies the geometric aspects???
> of the morphism f???. What did you have in mind for the ???we shall see later????
>
> Best wishes,
>
> Steve.
>
>

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