Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,394
questions
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"Noetherianess" of $\mathrm{Mod}(\mathbb{F}_{p,\square})$
In classical commutative ring theory it is quite immediate to see, that a field is noetherian in the following sense:
For any finitely generated $k$-vectorspace $M$, any sub-object is finitely ...
4
votes
0
answers
76
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Adjunction symbol
What are the reasons for the adjunction symbol $F\dashv G$ for a pair of functors $F:C\to D$ and $G:D\to C$? There is no explanation or motivation in the article of Kan where adjunctions are ...
7
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2
answers
168
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(When) does a morphism of monad induce adjoint functors between categories of algebras?
For monads $S$ and $T$ on a fixed Abelian category $C$, a morphism of monads $\sigma: S\rightarrow T$ induces a functor between Eilenberg-Moore categories $\sigma^*:C^T\rightarrow C^S$. This functor ...
7
votes
0
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147
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Morphisms in cube category $\Box$ = Compositions of morphisms in simplex category $\Delta$?
Let $\Delta$ be the simplex category. For $m \leq c \leq n$, let $[m] \to [c] \to [n]$ be the composition of two injective morphisms in $\Delta$.
We now define a category $\Box$ with same objects as $\...
5
votes
1
answer
142
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A pushout diagram of derived categories coming from an open cover of schemes
Suppose $X=U\cup V$ is the standard open cover of $X=\mathbb{P}^1$ by two affine lines. The descent theorems say that the diagram (with all arrows restriction maps)
$\require{AMScd}$
\begin{CD}
D(X) @&...
11
votes
0
answers
125
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Isbell duality for simplicial sets
$\newcommand{\IsbellSpec}{\mathsf{Spec}}\newcommand{\IsbellO}{\mathsf{O}}\newcommand{\Nat}{\mathrm{Nat}}$Isbell duality sets up an adjunction (see here for a short abstract summary)
$$\mathsf{O}\dashv\...
3
votes
1
answer
201
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Why is the universal $n$-gerbe universal? (HTT, 7.2.2.26)
Let $\mathcal{X}$ be an $\infty$-topos and let $A$ be an abelian group object of the category $\operatorname{Disc}(\mathcal{X})$ of discrete objects of $X$. Recall that a morphism $f:\widetilde{X}\to ...
2
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1
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112
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"$X$ is $n$-truncated $\iff$ $\Omega X$ is $(n-1)$-truncated" for connected pointed $X$. (HTT, 7.2.2.11)
In the proof of Lemma 7.2.2.11 of Higher Topos Theory, Lurie makes the following claim:
($\ast$) Let $n\geq1$ be an integer, let $\mathcal{X}$ be an $\infty$-topos, and let $1\to X$ be a pointed ...
4
votes
1
answer
116
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Cocompletion without cocontinuous functors
The forgetful functor from the 2-category $\mathsf{Cats}^{\mathrm{loc.small}}_{\mathrm{cocomp}}$ of locally small cocomplete categories and cocontinuous functors to the 2-category $\mathsf{Cats}^{\...
6
votes
1
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267
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Factoring through projective modules is an equivalence relation
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PHom{PHom}$I'm reading about stable module categories, and I have a question about the definition of the maps. Let $R$ be a ring, and take (left) ...
6
votes
1
answer
230
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Consequences of imposing conditions on the restricted Yoneda embedding of a functor
$\newcommand{\op}{\mathsf{op}}\newcommand{\yo}{よ}$Given a functor $F\colon\mathcal{C}\to\mathcal{D}$, the composition
$$\mathcal{D}\overset{\yo}{\hookrightarrow}\mathsf{PSh}(\mathcal{D})\xrightarrow{F^...
2
votes
1
answer
82
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Lax Gray tensor product and opposite categories:
For any two strict (infinity, infinity)-categories $A,B$ let $A \otimes B $ be the lax Gray tensor product of $A$ and $B$. Let $A^{op}$ be the opposite (infinity,infinity)-category, where morphisms in ...
1
vote
1
answer
132
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Are the minimal nondegenerate extensions universal?
We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...
4
votes
1
answer
271
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Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$ in a topos?
I am looking for ways to internalize the modal operator of necessity $\Box$, ending up with a morphism $\Box: \Omega \to \Omega$ satisfying the necessitation rule (if $\phi$, then $\Box \phi$) and the ...
5
votes
1
answer
336
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Endomorphisms of simple dualizable objects in a linear abelian monoidal categories
In When is an object in a linear or abelian category simple? Or: How should I define fusion categories? endomorphisms of simple objects in a k-linear abelian category are discussed. In the answer it ...
0
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0
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63
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Automorphism groups for simple objects in abelian linear categories
Let $\mathcal{A}$ be an abelian category that is also $k$-linear, where $k$ is some algebraically closed field. Let $X$ be a simple object in $\mathcal{A}$. What can we say about $\mathrm{Aut}(X)$? I ...
2
votes
1
answer
129
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In a monoidal category with duals is the coevaluation map determined by the evaluation?
For a monoidal category $(\mathcal{M},\otimes,1)$ and an object $X$ with a left dual $X^*$. Let $(e,c)$ be a pair of (co)evaluation maps for the pair $(X,X^*)$. Is it possible to have another map $c': ...
6
votes
1
answer
222
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About the characterization of categories of model of algebraic theories
So, in his Handbook of categorical algebra Vol 2, Borceux states a theorem (the 3.9.1, page 158) that says that:
Given a category $C$ with a functor $U:C \to Set$, $C$ is the category of models of a (...
2
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2
answers
173
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Boolean algebra object structure on coproduct of terminal object
I am asking for some clarification on this old question.
The context for my question is a cartesian closed category $ C $ with a binary coproduct and a terminal object $ I $. One of the answers claims ...
5
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0
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75
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Is the derived category of sheaves localised at pointwise homotopy equivalences locally small?
In order to define the cup and cross products in sheaf cohomology, Iversen makes computations in an intermediate derived category. If $K(X;k)$ is the triangulated category of cochain complexes of ...
5
votes
0
answers
48
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Is a pseudomonic and pseudoepic functor necessarily an equivalence of categories?
$\newcommand{\id}{\mathrm{id}}$Pseudomonic and pseudoepic functors form an appropriate 2-dimensional generalisation of monomorphisms and epimorphisms to categories.
Namely, a functor $F\colon\mathcal{...
1
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0
answers
108
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Vertex cleaving and edge contraction as graph morphisms
In some approaches to operads and properads, categories of trees and graphs are used as "indexing categories" for this structure (see, for instance, https://arxiv.org/abs/0902.1954 or https:/...
0
votes
1
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142
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Hopf algebras actions
Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions?
There must be a common core, if the same term is ...
1
vote
2
answers
433
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What is a cogroup and what are coactions?
What is a cogroup and what are coactions?
A very nice way to think about a group action on an object $X$ is as a group homomorphism from $G$ to $\operatorname{Aut}(X)$. Is there something similar for ...
3
votes
2
answers
178
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The evaluation and coevaluation maps for an object isomorphic to a dualisable object
Let $X$ be an object in a monoidal category with dual $X^*$ and evaluation and coevaluation maps $e$ and $c$. Now if we have an isomorphism $\sigma:X \to Y$, for some other object $Y$, then $Y$ must ...
7
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0
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173
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Can Postnikov towers converge without Postnikov completeness?
In Higher Topos Theory, Lurie says that "Postnikov towers are convergent" in a presentable $\infty$-category $\mathcal{C}$ if $\mathcal{C}$ is equivalent to the $\infty$-category $\mathrm{...
10
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0
answers
149
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Characterisations of functors $F$ such that $F^*$ or $F_*$ is [property], e.g. faithful, conservative, etc
I'm currently writing a comprehensive/encyclopedic set of notes on category theory, and one of the things I'm trying to do is gather all sorts of statements of the form
Let $F\colon\mathcal{C}\to\...
2
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0
answers
51
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Looking for a nice characterisation of functors $F$ whose precomposition functor $F^*$ is full
$\newcommand{\Obj}{\mathrm{Obj}}$Adámek–Bashir–Sobral–Velebil's On Functors Which Are Lax Epimorphisms proves the following result:
Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor. The following ...
5
votes
0
answers
98
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Are there exotic examples of a Lie group up to coherent isotopy?
This question is based on attempting to construct the (homotopy type) of Lie groups using Cobordism Hypothesis style abstract nonsense.
There is an $\infty$-groupoid of smooth, framed manifolds where ...
1
vote
0
answers
74
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Spans and multisets
For some time now, I have been trying to draw a formal relationship between spans and multisets. I have heard "tales" of how this might work, but no references. Here is what I have come up ...
3
votes
1
answer
225
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Grothendieck group and an almost localization
Let $T$ be a small triangulated category and let $S\subset T$ be a full triangulated subcategory. We denote this embedding by $I: S\rightarrow T$.
Let $F: T\rightarrow S$ be a triangulated functor ...
4
votes
0
answers
107
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Preservation of Kan extensions
I am currently studying the theory of kan extensions more seriously, but I'm surprised of the apparent absence if theorems of preservation/reflection. What I have in mind is something along the lines ...
13
votes
1
answer
389
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On Lemma 5.5.16 of Cisinski's "Higher Categories and Homotopical Algebra"
I have a question regarding Section 5 of Cisinski's
"Higher Categories and Homotopical Algebra".
Let us write $\mathbf{sSet}$ and $\mathbf{bisSet}$ for the categories of
simplicial sets and ...
6
votes
0
answers
111
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Points of the sheaf topos over Blass' category
There is a site $\textsf{Blass}$ used for (constructive) non-standard analysis, whose objects are sets equipped with a filter, and morphisms are continuous functions defined up to a small set. (It is ...
5
votes
1
answer
159
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Proof that the inclusion $\Delta \to \mathbf{Pos}$ preserves colimits
I am trying to to prove that the inclusion $\Delta \to \mathbf{Pos}$ preserves colimits, where $\mathbf{Pos}$ is the category of partially ordered sets and monotone map, and $\Delta$ is the full ...
8
votes
1
answer
509
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Is there a better name for the "Mayer-Vietoris Octahedral axiom" and has it been studied?
$\newcommand{\K}{\mathcal{K}}$Say $\K$ is a triangulated category with suspension $\Sigma:\K\simeq \K$. In Iversen's book "Cohomology of Sheaves", he doesn't exactly examine triangulated ...
2
votes
1
answer
82
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Godement product of lax 2-natural transformations
Which of the two obvious choices for the Godement product of lax $2$-natural transformations is ‘correct’?
Specifically, recall that for natural categories, functors and natural transformations as ...
4
votes
1
answer
98
views
What are the 2-categorical mono/epimorphisms in the 2-category of relations?
$\newcommand{\procirc}{\mathbin{\diamond}}\newcommand{\rightproarrow}{\mathrel{\rightarrow\mkern-17mu|\mkern7mu}}$The monomorphisms in the 1-category $\mathsf{Rel}$ of sets and relations are precisely ...
0
votes
1
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173
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Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically? [closed]
I asked this question on MSE here
This question was inspired by: The influence of conjugacy class sizes on the
structure of finite groups.
My question is as follows: Is there a way to study the ...
9
votes
1
answer
279
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$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$
This is a crosspost (with minor alterations).
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category ...
7
votes
1
answer
91
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Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces
From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric ...
6
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0
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131
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Does the localization functor $\mathcal{C}\to S^{-1}\mathcal{C}$ preserve finite colimits when $\mathcal{C}$ is not small? (size issues in proof)
$\def\colim{\operatorname{colim}}
\def\hom{\operatorname{Hom}}$Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms of $\mathcal{C}$ that is a left multiplicative system. The ...
5
votes
1
answer
254
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Quotients in categories of metric spaces
There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
3
votes
2
answers
248
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Directed colimit of fully faithful functors
Suppose that for every $n\in\mathbb{N}$ we have a category $\mathcal{C}_n$ and a fully faithful functor $F_n:\mathcal{C}_n\hookrightarrow \mathcal{C}_{n+1}$. My question is whether fully faithful ...
0
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0
answers
74
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Density of universe lifting functor in type theory
For some context, this is part of a larger story on relative monads, I have asked about a generalization of the Topos of coalgebras construction to the relative case here.
The linked proposition 2.5 ...
5
votes
1
answer
398
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Does coproduct preserve cohomology in differential graded algebra category
Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...
2
votes
0
answers
76
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Implementation of the nerve of a category in GAP
I am trying to compute the second cohomology group of the coset poset $$\mathcal{C}_G\mathcal{F}=\{gA\mid g\in G,A\in\mathcal{F}\}$$ for a finite group $G$ and a family $\mathcal{F}$ of subgroups of $...
3
votes
0
answers
135
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Is taking Freyd envelopes adjoint to taking stable module categories?
Let $T$ be an (idempotent-complete) triangulated category. Then the Freyd envelope $mod(T)$ is an abelian category, the universal recipient of a homological functor $T \to mod(T)$. The Freyd envelope ...
2
votes
1
answer
160
views
Factorization systems for vector bundles
Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
4
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0
answers
217
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Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?
In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension".
From [DLN, Theorem II (iii)], where the ...