"Variable and fixed" in categories

Question

We often find in Grothendieck terminology the words variable and fixed (or absolute).

For example in SGA 4 studies variable topological spaces, groups, and categories as examples of morphisms of topos.

I would like to know if there is a precise meaning of these two words in Grothendieck's terminology. Especially in the case of categories.

See for instance SGA 4 IV 4.6 "Le topos ̂ 𝐶 pour 𝐶 catégorie variable."

Maybe this is related to the relative point of view.

I think the words are being used more or less in their ordinary, everyday meaning. For example, the section about "$\hat{C}$ for variable $C$" discusses the presheaf topos $\hat{C}$ and morphisms of presheaf toposes $\hat{C} \to \hat{C}'$. — Zhen Lin, Commented Sep 22, 2022 at 0:51

Mozibur Ullah · Accepted Answer · 2022-09-22 02:50:31Z

1

I think this is really informal usage rather than formal usage.

By the way, the "relative" point of view can mean several things, for example:

transforming a stagement say about algebras to a functorial statement in $\mathit{Alg}$.
transforming a statement about a category to a slice category. For example, we can ask whether a Category is Cartesian Closed or whether all its slices are, in which case this is called locally Cartesian Closed.

answered Sep 22, 2022 at 1:21

Mozibur Ullah

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$\begingroup$ Even if I find interesting the examples of relativity, the question asks for clarification of variable categories. $\endgroup$
– user234212323
Commented Sep 22, 2022 at 19:18

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