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16
votes
0answers
400 views

History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
10
votes
0answers
371 views

Where is the representability of the moduli of curves with framed points proved?

There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of ...
7
votes
0answers
131 views

Yoneda embedding and Horn sentences

The following is taken from Borceux and Bourn's Mal'cev, Protomodular, Homological, and Semi-Abelian Categories. Metatheorem 0.1.3. Let $\mathcal P$ be a statement of the form $\varphi\implies \...
3
votes
0answers
154 views

Examples of Sheafification via Hypercovers

For a presheaf $F$ on a category equipped with a pretopology, one has the sheafification $F^{\sharp}$ of $F$. I know well the plus-construction of sheafification, which is presented in Artin's paper "...
2
votes
0answers
212 views

Two functorial definitions of schemes

I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way: Equip the category $\textbf {Psh}=\operatorname{Fun}(\...
1
vote
0answers
60 views

A question on uniformly corepresented functor

Let $\mathcal{F}$ be a functor from the category of $k$-schemes to sets, uniformly corepresented by $M$. Suppose $U$ is an open subscheme of $M$. I could not find a good reference for uniformly ...
1
vote
0answers
128 views

How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation?

How to determine a functor (natrually arising from geometry or homological algebra) to be locally of finite presentation? Is there any reference for such staff? My example of functors underlying this ...
0
votes
0answers
282 views

restriction and pullback of representable etale sheaf along closed immersion

I find that the restriction and pullback of representable etale sheaf along closed immersion are very confusing. I think they are different in general, I hope some experts can confirm my understanding ...