Questions tagged [abstract-nonsense]

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

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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 ...
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2 votes
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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 ...
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13 votes
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Is there a concrete application of topos theory?

The notion of topos was originally formulated in SGA 4 in the context of attacking the Weil conjectures. This formalism turned out to be unnecessary for the purposes of proving those conjectures. But ...
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4 votes
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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 $...
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7 votes
1 answer
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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 ...
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Big etale topos vs small etale topos

Are they equivalent? That is, given a sheaf of sets $\mathscr{F}$ defined on the small etale site on $X$, is there an essentially unique way to extend it to a sheaf on the big etale site on $X$? If ...
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18 votes
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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 ...
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3 votes
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$H$-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 ...
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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 ...
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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 ...
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6 votes
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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 ...
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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 ...
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3 answers
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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 ...
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2 votes
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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 ...
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1 answer
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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^...
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14 votes
2 answers
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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 ...
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11 votes
1 answer
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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) ...
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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 ...
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10 votes
2 answers
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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 ...
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7 votes
4 answers
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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 ...
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5 votes
1 answer
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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'&...
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8 votes
2 answers
732 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 ...
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2 votes
1 answer
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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 ...
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