11
votes
Accepted
Algebraic cycles, Chow spaces and Hilbert-Chow morphisms
Mathoverflow answer
In my thesis [R4], I gave an ad hoc definition of a Chow functor ($\mathrm{Chow}_r$ above) that was meaningful also in characteristic p and close to Barlet's and Angéniol's ...
- 5,039
10
votes
Reference for the Brown representability theorem in the case of locally presentable (∞,1)-categories
The theorem is, in fact, false even for finitely locally presentable $\infty$-categories, and indeed for the $\infty$-category of spaces, as has been known since Heller's paper
On the ...
- 3,411
9
votes
Condition for an equivalence of functor categories to imply an equivalence of categories
You can find this theoerm as Proposition 5.28 in Kelly's Basic concepts of enriched category theory. I guess the only extra condition you need is that the base for enrichment forms a topos. They are ...
- 1,482
8
votes
Accepted
Is every petite category essentially small?
It seems to me that the following are equivalent for a locally small category $B$:
$B$ is petite
every presheaf $q\colon B^{\mathrm{op}} \to \mathsf{Set}$ is small
for every presheaf $q$ and ...
- 383
8
votes
Condition for an equivalence of functor categories to imply an equivalence of categories
As AT0 said, there is an analogous theorem in Kelly's book for categories enriched over any base. So this will apply as soon as $C$ and $D$ are enriched over $\mathcal{S}$. But if $\mathcal{S}$ is a ...
- 66.7k
8
votes
Accepted
Yoneda map for a composition of a representable functor and an arbitrary functor
This property of the functor $T$ is called being a dense functor, or "densely generating functor". The notion was first introduced by Isbell in the case where $T$ is fully faithful under the ...
- 63.9k
7
votes
About the Yoneda objects of a locally presentable category
These objects are usually called absolutely presentable. In $Set^\mathcal A$, they are precisely retracts of representables. Presheaf categories are characterized as cocomplete categories having a ...
- 2,160
7
votes
Accepted
About the Yoneda objects of a locally presentable category
The usual name for "Yoneda objects" is "tiny" or "small-projective".
In general the tiny objects in a presheaf category are the retracts of representables. In particular, if $A$ is Cauchy-complete, ...
- 66.7k
7
votes
Accepted
Why does every chain complex have a map into its cone?
This is really just rephrasing Simon Henry's comment, but there is a natural transformation $F\to\text{Hom}(C,-)$ given by $(f,s)\mapsto f$, and so if $\text{Cone}(C)$ represents $F$ then by Yoneda's ...
- 35.2k
7
votes
Can the functor of the points of a scheme be characterized by its values on subcategories of the affine schemes?
For Noetherian complete local rings with a fixed residue field, any map to a scheme factors through the spectrum of the completed local ring at some point defined over that field. So the functor from ...
- 148k
6
votes
Accepted
Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?
Consider the terminal object $1\in\mathbf{Pos}$.
Then the closure of $1$ in $\mathbf{Pos}$ under all weighted colimits is $\mathbf{Pos}$ itself: If $X\in\mathbf{Pos}$, then $X\cong X\cdot 1$ is the ...
- 499
6
votes
Accepted
Is this functor $\mathcal{F}: \text{Sch}/\mathbb{Q}\longrightarrow \text{Sets}$ a sheaf?
No, this isn't true. Take $X = \mathbb{P}^1$ with $U_1$ and $U_2$ being the standard affine cover of $\mathbb{P}^1$. Let the coordinate rings of $U_1$ and $U_2$ be $k[t]$ and $k[t^{-1}]$, so the ...
- 156k
5
votes
Accepted
Any exact faithful functor is represented by a unique projective generator
Let $\mathcal C$ be the category of finite dimensional left modules over a finite dimensional ring $R$. Let $G: \mathcal C \to \mathrm{Vec}$ be an exact and faithful functor to finite dimensional ...
- 3,928
5
votes
Does every functor between Grothendieck categories have adjoints?
A Grothendieck category is locally (finitely) presentable; see for example Theorem 2.2 in this paper by Positselski and Rosický. And any colimit-preserving functor between locally presentable ...
- 53.3k
5
votes
Accepted
Why does the variety of ideals in this quaternion type algebra have a non-reduced structure?
It is not immediately obvious (at least to me) but it is an easy calculation. The following is too long for a comment, so I post it here:
Here is a simpler example of the same phenomenon: Let $A=\...
- 4,540
4
votes
Accepted
Why can I take the quotient of a relative elliptic curve by a finite locally free subgroup?
A reference for 1. is https://stacks.math.columbia.edu/tag/07S7
We can then answer question 2. like this:
The quotient morphism $E\to E/C$ is faithfully flat (this is a part of the conclusion of the ...
- 7,377
4
votes
About the Yoneda objects of a locally presentable category
Question 1: The Yoneda objects are precisely retracts of representables. There are quite a few buzzwords which are relevant here (Cauchy completion, idempotent splitting completion, Karoubi envelope, ...
- 53.3k
4
votes
Reference for the Brown representability theorem in the case of locally presentable (∞,1)-categories
(The following doesn't address the precise version of Brown representability asked about in the question body, but it does address some version of the title question, so I think for searchability it ...
3
votes
Reference for the Brown representability theorem in the case of locally presentable (∞,1)-categories
As pointed out to me by George Raptis, a detailed treatment of Brown representability for $(n,1)$-categories ($1≤n≤∞$), stable or not, is now available in
Hoang Kim Nguyen, George Raptis, Christoph ...
- 37.8k
3
votes
Accepted
Representability of Grassmannian functor by a scheme
First of all, being quasicompact is not a "strong finiteness assumption", come on :). For what you're doing you're actually free to restrict $F$ to quasi-compact quasi-separated schemes, or even just ...
- 915
2
votes
Accepted
Yoneda lemma for one object categories
Let me write $\def\B{\mathbf{B}}\B G$ for what you call $\mathbb{G}$.
You can check that presheaves on $\B G$ are precisely the right $G$-sets: the unique point of $\B G$ is sent to some set $X$, and ...
- 627
2
votes
Accepted
Proving the representability of a functor that is covered by open subfunctors
I found another way to prove this Theorem, which then also shows that this map is injective. Consider two Zariski sheaves $F,G:\mathrm{Sch}^{\circ}\rightarrow\mathrm{Set}$ and for each sheaf a cover ...
- 215
2
votes
Accepted
(Pro-)representable functors and full subcategories in homotopy theory
This is a partial answer. Broadly speaking, representability theorems break down into two types. In both cases, the functor $F$ has to satisfy some exactness condition. For Freyd type theorems, $F$ ...
- 30.3k
1
vote
Accepted
Construct morphisms of schemes on level of associated functors
I can't speak for Matt Emerton specifically, but my understanding is that it is conventional to describe maps in terms of $k$ points in such a way that there is a clear extension of the definition to $...
- 396
1
vote
Is a group scheme determined by its category of representations?
N̶o̶.̶ (YES, the answer above is correct.) The following example shows we cannot drop that condition: Work out the example of $G = E$ an elliptic curve/$k$. (The Rep category is severely degenerate ...
- 639
Only top scored, non community-wiki answers of a minimum length are eligible