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Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

86 votes
1 answer
5k views

Are there non-scalar endomorphisms of the functor $V\mapsto V^{**}/V$?

Let $K$ be a field. Are there non-scalar endomorphisms of the endofunctor $$ V\mapsto V^{**}/V $$ of the category of $K$-vector spaces? I asked a related question on Mathematics Stackexchange, but …
Pierre-Yves Gaillard's user avatar
27 votes
2 answers
2k views

Is every commutative ring a limit of noetherian rings?

Edit of Feb. 14, 2019. After Laurent Moret-Bailly's accepted answer, only Questions 4 and 5 remain open. I don't care that much about Question 4, but I'm very curious about Question 5, which is Do …
Pierre-Yves Gaillard's user avatar
20 votes
1 answer
970 views

Example of an additive functor admitting no right derived functor

I asked the same question a week ago on Mathematics Stackexchange but got no answer. What would be a simple example of an additive functor $F:\mathcal C\to\mathcal C'$ of abelian categories such that …
Pierre-Yves Gaillard's user avatar
19 votes
2 answers
1k views

If $\mathcal C^{\mathcal C}$ is equivalent to $\mathcal C$, is $\mathcal C$ necessarily equi...

Let $\mathcal C$ be a category which is equivalent to the category $\mathcal C^{\mathcal C}$ of its endofunctors. Is $\mathcal C$ necessarily equivalent to a category having exactly one object and …
Pierre-Yves Gaillard's user avatar
17 votes
Accepted

"Sums-compact" objects = f.g. objects in categories of modules?

It seems to me the references in this Mathematics - Stack Exchange answer contain the requested information. EDIT 1. Here is an excerpt from Hyman Bass's book Algebraic K-Theory, W. A. Benjamin (196 …
Pierre-Yves Gaillard's user avatar
14 votes
2 answers
669 views

$\mathcal A^{\mathcal A}\sim\mathcal B^{\mathcal B}\implies\mathcal A\sim\mathcal B\ ?$ (Doe...

I asked this question on Mathematics Stackexchange, but got no answer. Let $\mathcal A$ and $\mathcal B$ be nonempty categories whose categories $\mathcal A^{\mathcal A}$ and $\mathcal B^{\mathcal B} …
Pierre-Yves Gaillard's user avatar
13 votes
0 answers
295 views

Is $\mathrm{Hom}(P^i,P^j)$ a finite set? ($P=$ power set functor, $i\equiv j\bmod2$)

Let $P:\textbf{Set}\to\textbf{Set}$ be the contravariant power set functor, and put $P^n:=P\circ\cdots\circ P$ ($n$ factors), so that $P^n$ is a covariant (resp. contravariant) endofunctor of $\textbf …
Pierre-Yves Gaillard's user avatar
13 votes
1 answer
465 views

Does the Cantor-Schröder-Bernstein Theorem hold in the category opposite to the category of ...

I asked this question on Mathematics Stackexchange, but got no answer. Let $A$ and $B$ be noetherian commutative rings with one, and let $f:A\to B$ and $g:B\to A$ be epimorphisms. Are the rings …
Pierre-Yves Gaillard's user avatar
11 votes
2 answers
656 views

Equivalence of the definitions of a sheaf in SGA4 and in "Categories and Sheaves"

I asked this question on Mathematics Stack Exchange, but got no answer. I don't understand why the definition of a sheaf (Definition 17.3.1 (ii)) given in the book [KS] Categories and Sheaves by Kashi …
Pierre-Yves Gaillard's user avatar
10 votes
1 answer
260 views

Is $\operatorname{Hom}(F,G)$ finite if $F$ and $G$ are endofunctors of the category of finit...

I asked this question on Mathematics Stackexchange but got no answer. Are there endofunctors $F$ and $G$ of the category of finite sets such that there are infinitely many natural transformations fro …
Pierre-Yves Gaillard's user avatar
9 votes
1 answer
818 views

Are epimorphic endomorphisms of noetherian commutative rings always injective?

This question was asked, but not answered, on Mathematics Stackexchange. [In this post "ring" means "commutative ring with one".] Let $A$ be a noetherian ring, and let $f:A\to A$ be an endomorphism …
Pierre-Yves Gaillard's user avatar
5 votes
1 answer
177 views

Comparing $\mathcal C$ and $\mathcal C^{\mathcal C}$ (where $\mathcal C$ is a category)

This is a followup to this question. (Matt Feller also mentioned this followup in a comment to the question linked to above.) For any category $\mathcal C$ write $\mathcal C^{\mathcal C}$ for the cat …
Pierre-Yves Gaillard's user avatar
5 votes
Accepted

Comparing $\mathcal C$ and $\mathcal C^{\mathcal C}$ (where $\mathcal C$ is a category)

If $\mathcal C$ is the category attached to the ordered set $(\mathbb R,\le)$, then $[\operatorname{Ob}(\mathcal C)]$ coincides with the set $\mathbb R$ and $[\operatorname{Ob}(\mathcal C^{\mathcal C} …
Pierre-Yves Gaillard's user avatar
5 votes

If $\mathcal C^{\mathcal C}$ is equivalent to $\mathcal C$, is $\mathcal C$ necessarily equi...

This is a partial answer. I tried to mimic the proof of Theorem 3 in [1] Complete lattices and the generalized Cantor theorem by Roy O. Davies, Allan Hayes and George Rousseau, published in Proc. Am …
Pierre-Yves Gaillard's user avatar
5 votes

Localizing an arbitrary additive category

Here is the statement of Exercise 8.4 p. 202 of [KS] Categories and Sheaves by Kashiwara and Schapira: (a) Let $\mathcal C$ be an additive category and $\mathcal S$ a right multiplicative system …
Pierre-Yves Gaillard's user avatar

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