Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,364
questions
4
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Is there a category theoretic definition of a cryptographic commitment scheme?
I'm trying to come up with a composeable framework for cryptographic commitment schemes, where inclusion proofs can be combined in different ways. I'm thinking this can be done with category theory, ...
2
votes
0
answers
139
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Infinitesimal criteria for unramified morphism on stacks
In an Artin stack, the tangent complex is fundamentally a derived object, often viewed as a 2-term complex via some quotient presentation, as discussed in Raskin's note here, for example. In ...
2
votes
0
answers
141
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Trying to decode a module functor
This is section 5.1 from, ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten.
Let $A$ be a commutative unitary ring, let $R$ be a commutative monoid, and let $\alpha : R \to A$ be a ...
5
votes
1
answer
307
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A result on symmetric closed monoidal categories
In this discussion from the categories mailing there is mention of the following result by Robin Houston, supposedly proved in 2006:
Theorem. Let $\mathcal{C}$ be a symmetric closed monoidal category,...
6
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1
answer
596
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When is the cofibrant replacement of a product the product of the cofibrant replacements?
I'm in a situation where I'd like to prove $Q(E\otimes E) \simeq QE \otimes QE$ for a monoid $E$ in a symmetric monoidal model category. I know it's not true in general that $Q(E\otimes F)\simeq QE \...
8
votes
0
answers
124
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Locally presentable and accessible categories without the axiom of choice?
Is there a good reference for the study of locally presentable and accessible categories without the axiom of choice? For instance, it seems one will need to understand:
What is a good notion of $\...
5
votes
1
answer
181
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3-functoriality of the lax Gray tensor product
In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For ...
1
vote
0
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194
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Kan extensions in Grothendieck school
Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early ...
3
votes
1
answer
141
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What are the internal adjunctions in the bicategory $\mathsf{Span}$?
Recently I've been trying to understand spans better, in particular how they relate to relations, as both may be thought of as "multivalued functions between sets" (see Bruni and Gadducci - ...
106
votes
15
answers
35k
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Most striking applications of category theory?
What are the most striking applications of category theory? I'm trying to motivate deeper study of category theory and I have only come across the following significant examples:
Joyal's ...
2
votes
1
answer
113
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Duality in a monoidal category as a functor
In a rigid monoidal category $\mathcal{M}$ every object has a (say left) dual. Is the process of taking duals functorial? More specifically - is there a well-defined functor
$$
\mathcal{M} \to \...
4
votes
0
answers
535
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$\mathbb{Z}[T]$-Solidification in light condensed setting
In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...
3
votes
1
answer
250
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Do objects in the derived category behave stackily?
It is well known that derived categories (I'm particularly thinking of constructible derived categories and derived categories of D-modules) don't form a stack. In particular given morphisms in the ...
4
votes
1
answer
248
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Interesting Grothendieck topologies or coverages on the category Prob
I am currently trying to understand Grothendieck Topologies and coverages and want to endow the category Prob, consisting of finite probability spaces and measure preserving maps, with a Grothendieck ...
-1
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1
answer
155
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Does a symmetric monoidal functor between cartesian monoidal categories automatically preserve products? [closed]
Suppose $\cal A, B$ are cartesian monoidal categories, that is, categories equipped with a choice of finite products (including the nullary one, $1$). Suppose, moreover, that $F: \cal A \to B$ is a ...
5
votes
0
answers
113
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Gauge Lie groupoid associated to $SO(3)$ double cover
From each Lie group $G$ and principal $G$-bundle $P \rightarrow E$ one can form an associated (or gauge) Lie groupoid as the quotient of pair groupoid by the action of $G$ on $P \times P$
$$ \frac{P \...
3
votes
1
answer
182
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Transitivity axiom for a Grothendieck Topology
I am currently trying to define a Grothendieck Topology on the category Prob which consists of finite probability spaces with measure preserving maps between them.
I declared the covering sieves of an ...
2
votes
0
answers
128
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Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces
I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
1
vote
1
answer
361
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Why is "everything staying correct" for simplicial spaces?
I recently need a simplicial generalization of some theorem for rigid spaces, namely Theorem A holds for a rigid space $X$ and I want a Theorem $A_\bullet$ for a simplicial rigid space $X_\bullet$. ...
2
votes
1
answer
81
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Are the injections of a coproduct a cover in the canonical pretopology?
Assume we're in a category $C$ with all pullbacks and finite coproducts.
Recall that the canonical coverage of $C$ is the finest Grothendieck (pre) topology for which all representables are sheaves. A ...
5
votes
0
answers
145
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In what algebraic categories do finitely presentable objects form a dense cogenerator?
For each $C$ locally finitely presentable category, the full subcategory of finitely presentable objects $C_{fp}$ is a dense generator, i.e. the natural functor $C \to \mathrm{PSh}(C_{fp})$ is a full ...
8
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0
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219
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What is the exact definition of the $\infty$-topos of sheaves on a localic $\infty$-groupoid?
The category $\mathrm{Locale}$ is equivalent to the category $0\text{-}\mathrm{Topos}$ .
The 2-category $\mathrm{LocalicGroupoid}$ (with suitable localization) is equivalent to the 2-category $1\text{...
3
votes
1
answer
146
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Generalization of category algebra
Let $R$ be a commutative ring. Let $\mathcal C$ be a category that has finitely many objects. The category algebra $R[\mathcal C]$ of $\mathcal C$ consists of finite sums $\sum a_i f_i$, where $f_i$ ...
6
votes
1
answer
195
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In a weak factorization system, the left class is left cancellative iff the right class is what?
Let $\mathcal K$ be a locally presentable category, and let $(\mathcal L, \mathcal R)$ be a cofibrantly-generated weak factorization system thereon. We say that $\mathcal L$ is left cancellative if $...
1
vote
0
answers
104
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Lengths and additive invariants which preserve positivity
The length of a module is well-known to be an additive invariant of finite-length modules. That is, if $R$ is a ring and $Art(R)$ its category of finite-length modules, then $length : Ob (Art(R)) \to \...
0
votes
0
answers
91
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Can 2 coverages generate the same Grothendieck Topology if the category is large?
I am currently analyzing a category which is not small, but locally small. I have seen that any coverage on any small category $\mathcal{C}$ generates a unique Grothendieck Topology on $\mathcal{C}$ ...
2
votes
1
answer
221
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Comparing the exit path category and the nerve of a stratified space
Let $P$ be a finite poset, and $X$ be a topological space stratified by $P$, in the sense that $X$ is equipped with a continuous map $X \to P$ in the Alexandroff topology (or equivalently with a ...
23
votes
1
answer
2k
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Condensed vs pyknotic vs consequential
As is probably clear from my previous questions, I am coming to "condensed mathematics" from the naive perspective of a category theorist, without much knowledge of the intended applications ...
10
votes
0
answers
494
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Isbell duality between algebras and sheaves
nLab says on Isbell duality, the following:
A general abstract adjunction
$(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves} \leftrightarrows \mathrm{Presheaves}$
relates (higher) ...
5
votes
1
answer
252
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Do finitely presentable $\infty$-groupoids precisely correspond to the finite cell complexes?
In the Higher Topos Theory, Example 1.2.14.2 says “finitely presentable $\infty$-groupoids correspond precisely to the finite cell complexes” But, for example, $K(\mathbb{Z}, 2)$ is seems finitely ...
9
votes
2
answers
588
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What algebraic structure controls endomorphisms of algebras over a Lawvere theory
Given a Lawvere theory $T,$ is it possible to describe a Lawvere theory $\textrm{End}(T)$ such that $\textrm{End}(T)$-algebras describe "endomorphisms of $T$-algebras"?
In other words, what ...
6
votes
0
answers
216
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Generalizing uniform structures as Grothendieck topologies
Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
7
votes
1
answer
142
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The change-of-monoid adjunction between categories of modules induced by a morphism of monoids
Let $\mathcal{M}$ be a cocomplete closed symmetric monoidal category. Let $A, B$ be monoids in $\mathcal{M}$ and $f: A \rightarrow B$ be a morphism of monoids. The morphism $f$ induces the extension ...
12
votes
2
answers
705
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Examples of non-polynomial comonads on Set?
Question: What are examples of comonads on $\mathbf{Set}$ that are not polynomial?
Background: polynomial functors and comonads on Set
A functor $F\colon\mathbf{Set}\to\mathbf{Set}$ is called ...
3
votes
0
answers
302
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What is known about representations of $S_n$ in other categories?
Is anything known about representations of the symmetric group $S_n$ for categories other than $\textbf{Vect}_k$, vector spaces and linear maps over a field $k$.
That is, a group $G$ can be considered ...
29
votes
3
answers
4k
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What is the precise relationship between pyknoticity and cohesiveness?
Pyknotic and condensed sets have been introduced recently as a convenient framework for working with topological rings/algebras/groups/modules/etc. Recently there has been much (justified) excitement ...
11
votes
2
answers
695
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The convolution of comonads is a comonad
$\def\Cat{\mathbf{Cat}}\def\Set{\mathbf{Set}}\def\A{\mathcal{A}}$I stumbled into the following statement:
Let $\Cat(\Set,\Set)_s$ be the category of small functors[¹] $F : \Set \to\Set$ and let $F,G$ ...
13
votes
2
answers
486
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How many automorphisms are there of the category of filtered spectra?
Dold-Kan type theorems tell us that lots of categories are Morita-equivalent to the simplex category $\Delta$. In other words, there are a lot of stable $\infty$-categories which are secretly ...
6
votes
1
answer
726
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The "binary" product preserves pushouts?
In the category Set of sets and functions, consider the functor F(X) = X * X where * is the product (its action on arrows is just F(f) = f * f). Does this functor preserve pushouts? Or at least ...
4
votes
0
answers
111
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Cocartesian fibration classifying $\mathrm{Fun}(F,G)$
Throughout this question we consider $\infty$-categories.
Fix a cartesian fibration $p : \mathcal{F} \to \mathcal{C}$ and a cocartesian fibration $q : \mathcal{G} \to \mathcal{C}$ which straighten to $...
3
votes
0
answers
117
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proper smooth dg-categories and colimit
Let $I$ be a filtered category and $\{k_i\}_{i\in I}$ be a system of commutative rings over $I$. Toen proved that there is an equivalence of categories
$$
\text{Colim}-\otimes^{\mathbb{L}}_{k_i} k:\...
2
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0
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217
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Quantum scattering experiments: C-modules, N-modules and their monads
I am working on a theory of particle physics where we use monads. I have a few conjectures that I need to check.
The category of $\mathbb{C}$-modules is monadic over set
The category of $\mathbb{N}$...
4
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0
answers
117
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Is there a faithful functor from the freely generated bicartesian closed category to $\mathbf{Set}$?
Does there exist a faithful (bicartesian closed) functor $\operatorname F$ from the freely generated bicartesian closed category $\mathbf B$ to $\mathbf{Set}$? Preferably, $\mathbf B$ should contain ...
19
votes
3
answers
1k
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$\operatorname{Ind}(C^I) = \operatorname{Ind}(C)^I$?
$\DeclareMathOperator\Ind{Ind}$Let $C$ be a (small) category, and $I$ a finite category, is it true that the natural functor $\Ind(C^I) \to \Ind(C)^I$ is an equivalence of categories? where $\Ind$ ...
11
votes
1
answer
1k
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Existence of skeletons in ZFC
Influenced by this question from a fellow lagomorph, I would like to get to the bottom of existence of a skeleton of a category. I want to stay in ZFC, so I do not assume the global axiom of choice. ...
3
votes
1
answer
85
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How to represent morphisms in a fibration in the internal type theory
Given a fibration $p:\mathcal{E \to B}$, we can work with a minimal type theory with semantics in $p:\mathcal{E \to B}$, its internal type theory.
The type theory for $p$ is dependent, with contexts ...
3
votes
1
answer
515
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Do CGWH spaces form an exponential ideal in Condensed Sets?
If $X$ is any condensed set and $Y$ is a compactly generated weak Hausdorff (CGWH) space (a.k.a. $k$-Hausdorff $k$-space), is $Y^X$ again a CGWH space? To be more precise, is $(\:\underline{Y}\,)^X$ ...
1
vote
3
answers
436
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Smooth affine algebras are Calabi-Yau
Are all smooth affine algebras over a field Calabi-Yau?
I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/
1
vote
1
answer
195
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Proof in Higher Algebra that $\mathcal{C}at(\mathcal{K})$ is presentable
In Higher Algebra Lemma 4.8.4.2, Lurie shows that for $\mathcal{K}$ a small set of simplicial sets, the $\infty$-category $\mathcal{C}at(\mathcal{K})$ of small $\infty$-categories with $K$-shaped ...
38
votes
7
answers
8k
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What's a reasonable category that is not locally small?
Recall that a category C is small if the class of its morphisms is a set; otherwise, it is large. One of many examples of a large category is Set, for Russell's paradox reasons. A category C is ...