Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [abstract-nonsense]

Arguments working entirely at a high level of abstraction, particularly category-theoretic arguments.

3
votes
1answer
76 views

$H$-space structure on coloured algebras

If $\mathcal{O}$ is a (classical) topological operad with unit $1\in \mathcal{O}(1)$, $\mathcal{O}(0)=\{0\}$ and multiplications $(m;a_1,\dotsc,a_r)\mapsto m(a_1,\dotsc,a_r)$. Let $X$ be an algebra ...
1
vote
0answers
52 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 ...
2
votes
2answers
326 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 ...
5
votes
1answer
357 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 ...
8
votes
4answers
376 views

Abstract treatment of multivariate calculus relevant for optimization

After studying the basics of (convex) optimization, I've become convinced there's sometimes a conceptual benefit in thinking of quantities like gradients etc. in a coordinate-free way, and keeping ...
3
votes
3answers
244 views

Ref request: making a homotopy equivalence fibrewise, in an abstract setting (eg fibration categories)

The following useful lemma holds in a variety of settings: Lemma. Let $p : Y_1 \to X$, $p_2 : Y_2 \to X$ be fibrations over a common base, and $f : Y_1 \to Y_2$ a map over $X$ that is a homotopy ...
2
votes
0answers
89 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
1answer
390 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^...
14
votes
2answers
1k 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 ...
11
votes
1answer
418 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
0answers
739 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 ...
8
votes
2answers
644 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 ...
7
votes
4answers
2k 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
1answer
366 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'&...
4
votes
2answers
516 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
1answer
491 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 ...