**2**

votes

**0**answers

75 views

### What should one “do” to “strictify” a triangle of transformations coming from a lax commutative triangle of functors?

I would like to apologize for this rather stupid abstract nonsense question.
Let $h=f\circ g$ for composable functors $f,g$; assume that there exist left or right adjoints to $f$ and $g$. Then it ...

**1**

vote

**1**answer

338 views

### Are these two “FUNCTORS” adjoint?

I am considering the following correspondence:
Let $X$ be quasi compact quasi separated schemes.Consider a pseudo functor \begin{equation}Sch\rightarrow CAT :U\mapsto Qcoh(U),f:U\rightarrow V\mapsto ...

**11**

votes

**2**answers

680 views

### If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?

The other direction is well known.
I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering ...

**0**

votes

**0**answers

54 views

### Change of relative base

If $k$ is a commutative ring, $A$ a $k$-algebra and $\phi: k \rightarrow k'$ is a morphism of rings then how (/under what conditions) can the relative homology functors $Ext_{A/k}(-,-)$ and ...

**4**

votes

**0**answers

238 views

### Why does this setting imply that a category is Grothendieck?

I came across the following Lemma in Mitsuyasu Hashimoto's Equivariant Twisted Inverses; it is Lemma 11.2 on page 107 of this pdf.
Let $\mathcal{A}$ be an abelian category which satisfies the (AB3) ...

**0**

votes

**0**answers

225 views

### Finite separable extension of fields imply the number of intermediate subfield is finite

The proof of statements either uses Galois theory or Artin primitive element theorem.I would like to know whether there is a proof without using these.The reason to avoid using Galois theory is that ...

**6**

votes

**2**answers

315 views

### Is there a measure / probability theory in a topos of “generalized measure spaces”?

Consider the category with the standard Lebesgue measure space $\Omega$ as its only object and measure type preserving nonincreasing (equivalence classes of) maps as morphisms.
Question: is there an ...

**4**

votes

**3**answers

887 views

### The ABC of categories: ABstract vs Concrete

From Wikipedia:
A concrete category is a category that is equipped
with a faithful functor to the category of sets.
This functor makes it
possible to think of the objects of the category ...

**4**

votes

**1**answer

314 views

### Are strict pushout squares in Cat exact squares?

Let $C,C',D\in \operatorname{Cat}$, and consider the following strict pushout square
$$\begin{matrix}
C&\overset{f}{\to} &D^{op}\\
\downarrow^\pi&\swarrow&\downarrow^{\iota_2}\\
...

**3**

votes

**2**answers

322 views

### The single-plus construction is not the left adjoint of the inclusion of separated presheaves?

Convention:
Before I start, please note that it is not going to be sufficient to assume that the topology is subcanonical or that the site $C$ has finite limits, since the application I have in mind ...

**2**

votes

**1**answer

463 views

### Abstract classes of anodyne maps (relative to an interval) in a presheaf category are stable under smash products with monomorphisms?

Let $A$ be a small category, and let $X:=Psh(A)$ denote the category of presheaves on $A$. It is a theorem that for any such category $X$, there exists a small set $M$ of monomorphisms admitting the ...