I believe that in his paper "Notions of topos" Ross Street used the name "V-moderate category" for this or a closely related notion. There the point was that V-moderate categories have another advantage over locally V-small ones, namely that their objects can (assuming the axiom of choice) be well-ordered with all initial segments being V-small. On Mon, Jan 1, 2018 at 5:10 AM, Paul Blain Levy <P.B.Levy@cs.bham.ac.uk> wrote:
Hi,
Let V be a Grothendieck universe.?? A "V-set" is an element of V, and a "V-class" is a subset of V.
Say that a category C is "V-included" when it has the following two properties.
(1) ob C is a V-class.
(2) C(x,y) is a V-set for all x,y in ob C.
The advantage of V-inclusion over local V-smallness (i.e. condition (2) alone) is that V-included categories are W-small for every universe W greater than V, whereas locally V-small categories are not, in general.
Furthermore, all the standard categories constructed from V are V-included.?? (Except for the ones that are not even locally V-small, like the category of V-included categories.)
Is there a standard name for V-inclusion?
Paul
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