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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