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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 …

Here is a proof of the "plausible claim" in Will Sawin's answer. The "plausible claim" is: "There are no nonzero canonical (i.e. functorial) elements of $V^{*a}\otimes V^{*b}$ for any $a,b$." Here we …

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 …

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 …

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 …

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 …

I asked a closely related question on Mathematics Stackexchange but got no answer. Let $\mathbf1$ be a category with exactly one object and one morphism, and, for any category $\mathcal C$, write $\ma …

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 …

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} …

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} …

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 …

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 …

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 …

This is a sequel to this question. Let $k$ be a field, let $A$ be the $k$-algebra $k[\varepsilon]$ with $\varepsilon^2=0$, and consider the following three abelian categories: $\bullet\ \text M(A)$ …

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 …