All Questions
Tagged with abstract-nonsense ct.category-theory
16 questions
2
votes
0
answers
215
views
Proving that the functor induced by some inclusion functor has a left adjoint
Let $\mathcal{A}$ be an abelian category and $\mathcal{B}\subseteq \operatorname{Proj}(\mathcal{A})$ be a full additive subcategory of $\mathcal{A}$. We define the full subcategory $\mathcal{B(A)}$ of ...
- 121
2
votes
0
answers
189
views
Relation between push forward by diagonal morphism and higher direct image functors
Let $f : X \to Y$ be a morphism between two Noetherian schemes. Then $f_*$(respect to $R^1f_*$) sends coherent sheaves to coherent sheaves if and only if $f$ is universally closed (respect to ...
- 121
2
votes
1
answer
232
views
Morphisms from the empty diagram
Let $X$ be an object in a category, and let $D$ be the empty diagram in the same category (containing no objects, and therefore no morphisms).
What should $\text{Hom}(D,X)$ be?
The only reasonable ...
- 4,164
4
votes
0
answers
319
views
Polynomial objects in any concrete category
EDIT: The original question had a trivial answer: it's just a coproduct. New question below
New Question: As shown below, in the category of commutative unital rings, the coproduct of a ring $R$ with $...
- 438
11
votes
1
answer
1k
views
Are categories special, foundationally?
Some folks over at nLab want to use categories as a foundation for all of mathematics, I'm guessing as an alternative to sets. Sets work fine, and so do categories, so I have started wondering what ...
- 237
18
votes
2
answers
744
views
Properties of categories that can not be proven by abstract nonsense
What are examples of properties of particular categories that can be formulated in categorical language and "feel" like they ought to be provable formally but they actually are not? I think that this ...
user137597
1
vote
0
answers
60
views
Comments/references on an obscure category of "rudimentary representations"
Consider a finite toset (cool word) $A=\{a_1<\ldots<a_N\}$. Let an ordered monomial be a finite formal product $b_1\ldots b_M$ with $b_i\in A$ and $b_i\le b_{i+1}$.
Consider the following ...
- 3,513
2
votes
2
answers
358
views
Detecting Universals
An operation on a category $C$ is a functor $$F: C^n \to C.$$
I'd like to detect those construction that are universal. Easiest example is coproduct, which is a colimit.
In this sense to be ...
- 9,111
6
votes
1
answer
566
views
Is there a theory of decomposition into indecomposables? What's the relation to idempotents?
Call a nonzero object of a pointed category simple if it has no proper quotients, and indecomposable if it's not the product of two objects (dual to connected).
Idempotents seem to pop up in many ...
- 10.5k
2
votes
0
answers
105
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 ...
- 16.9k
1
vote
1
answer
416
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 f^...
- 1,982
16
votes
2
answers
3k
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 ...
- 1,982
11
votes
1
answer
537
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) ...
- 251
7
votes
4
answers
3k
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 as sets with
...
- 9,720
5
votes
1
answer
480
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}\\
C'&...
- 19.6k
10
votes
2
answers
935
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 ...
- 19.6k