Questions tagged [ct.category-theory]

Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

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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 $X_\bullet$. ...
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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 \...
Alexander Golys's user avatar
8 votes
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219 views

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{...
Arshak Aivazian's user avatar
1 vote
0 answers
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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 \...
Tim Campion's user avatar
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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 $...
Tim Campion's user avatar
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3 votes
1 answer
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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 ...
Maat's user avatar
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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 $f_i$ ...
Hang's user avatar
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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 $\mathcal{C}$ ...
Maat's user avatar
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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 ...
Arshak Aivazian's user avatar
9 votes
2 answers
588 views

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 ...
Grisha Taroyan's user avatar
6 votes
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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 ...
Nik Bren's user avatar
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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 ...
Angelos's user avatar
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12 votes
2 answers
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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 ...
David Spivak's user avatar
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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 ...
Jackson Walters's user avatar
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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 $...
daniel gratzer's user avatar
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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:\...
OOOOOO's user avatar
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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 ...
Johan Thiborg-Ericson's user avatar
4 votes
1 answer
248 views

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 ...
Maat's user avatar
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11 votes
1 answer
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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. ...
Bugs Bunny's user avatar
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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 ...
Markus Zetto's user avatar
5 votes
1 answer
337 views

Does the category of integral domains admit a symmetric monoidal structure?

Let $\mathbf{Int}$ be the category of integral domains with injective homomorphisms. Does it admit a symmetric monoidal structure? If so, can we choose $\mathbb{Z}$ as the unit object? If it helps to ...
Martin Brandenburg's user avatar
6 votes
1 answer
283 views

Relationship between canonical topology on a topos and its site of definition

The canonical (Grothendieck) topology for a category $C$ is the largest (finest) topology such that every representable presheaf over $C$ is a sheaf. According to First Order Categorical Logic Lemma 1....
Joey Eremondi's user avatar
4 votes
1 answer
388 views

Does the rank of a subfunctor not exceed the rank of a functor?

It is known that Vopenka's principle is equivalent to the statement “a subfunctor of a accessible functor is accessible” (Adámek and Rosický, Cor 6.31 in Locally Presentable and Accessible Categories)....
Arshak Aivazian's user avatar
7 votes
0 answers
141 views

An abelian category with a full embedding from topological abelian groups

I know this is a very vague question, but I can't think of a better question to ask. Consider the category $\mathscr{C}$ of pro-sets created by diagrams containing only injections. The objects $A: \...
Charles Wang's user avatar
11 votes
1 answer
400 views

Is an exponentiable fibration with contractible fibers a homotopy equivalence?

Question: Let $p : E \to B$ be an exponentiable functor of $\infty$-categories. Suppose that for every $b \in B$, the geometric realization of the fiber $|p^{-1}(b)|$ is contractible. Then does $p$ ...
Tim Campion's user avatar
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3 votes
1 answer
85 views

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 ...
seldon's user avatar
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1 vote
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What are the morphisms in the category of retractions?

In Michael Shulman's Framed bicategories and monoidal fibrations Example 12.10 he defines a category $\operatorname{Retr}(\mathcal{C})$ as the "category of retractions in $\mathcal{C}$". He ...
Andrew's user avatar
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13 votes
2 answers
486 views

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 ...
Tim Campion's user avatar
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5 votes
2 answers
511 views

What is the intuitive difference between these two simplicial subdivision functors?

$\newcommand{\sset}{\mathsf{sSet}}\newcommand{\poset}{\mathsf{Poset}}\newcommand{\p}{\mathscr{P}^{\mathsf{nd}}}\newcommand{\N}{\mathcal{N}}\newcommand{\sd}{\operatorname{sd}}$Following this paper, I ...
FShrike's user avatar
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10 votes
2 answers
960 views

Why are the source-target rules of composition always strictly defined?

General categorical definitions always have two variants, a strict one, in which associativity and unity hold as equalities, and a weak one, in which they hold up to equivalence. However, every ...
Alexander Praehauser's user avatar
14 votes
1 answer
1k views

What is decategorification?

A decategorification is, roughly, some procedure $\Phi$ which inputs some sort of $n$-categorical data and outputs some sort of $(n-1)$-categorical data. Whatever this means, categorification is ...
Tim Campion's user avatar
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0 votes
1 answer
282 views

Can we generalise groupoids to monoid-oids? [closed]

Groups correspond to one object categories where every morphism is an isomorphism. Monoids correspond to one object categories. Groupoids correspond to small categories where every morphism is an ...
Diego de la Paz's user avatar
7 votes
0 answers
286 views

A system with distinct infinite cardinalities but no "best" version of $\mathbb{N}$

Let $\mathfrak{S}=(M_z,U_z)_{z\in\mathbb{Z}}$ be a sequence such that for each $z\in\mathbb{Z}$ we have $M_z\models\mathsf{ZFC}+$ "$U_z$ is a nonprincipal ultrafilter on $\omega$" (so in ...
Noah Schweber's user avatar
9 votes
1 answer
430 views

Is there a shape-independent definition of (∞,1)-categories?

For all definitions of $\infty$-categories I am aware of, an $(\infty,1)$-category is defined via reference to some shape, be it simplices in a form of a quasi-category or a cubical analogue of a ...
lemmanade's user avatar
6 votes
1 answer
379 views

A possible alternative model for $\infty$-groupoids

I've been studying homotopy theory on myself for quite some time now, and it is to my understanding that there's still no generally accepted definition for $\infty$-groupoids. The closest to this is ...
XiaohuWang's user avatar
5 votes
0 answers
125 views

Is the opposite of the category of $\kappa$-Lindelöf Hausdorff spaces locally presentable?

Gelfand duality tells us that the category of compact Hausdorff spaces (with continuous maps as morphisms) is contravariantly equivalent to the category of commutative, unital $C^\ast$-algebras (with $...
Tim Campion's user avatar
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2 votes
0 answers
348 views

What is the nerve of this category?

If $\mathcal{C}$ is a thin category, we call $U \subseteq \mathrm{Ob}(\mathcal{C})$ open if for every object $X \in U$ and any morphism $X \to Y$, we also have $Y \in U$. This declares an Alexandrov ...
Bastam Tajik's user avatar
12 votes
2 answers
713 views

Concrete representation of coend in linear algebra

$\require{AMScd}$Teaching coend calculus to a PhD student led me to this "elementary" computation that I would like to perform explicitly. Consider the functor $F : (\mathbb N,\le)^\text{op}\...
fosco's user avatar
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1 vote
1 answer
307 views

Is there anyway to formulate the Alexandrov topology algebraically?

One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set. Given this, one finds a one-to-one correspondence between ...
Bastam Tajik's user avatar
6 votes
2 answers
531 views

Examples of bilimits that aren't 2-limits, and some related questions

Recently I've received an email from Sori Lee about an earlier question I had asked, and we ended up with a number of questions about 2-limits and 2-bilimits which I couldn't quite answer, and decided ...
Emily's user avatar
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5 votes
2 answers
415 views

The category of groupoids vs the category of sets

Hopefully this question makes sense. As we know that Kan complexes are the "$\infty$-version" of groupoids for $\infty$-categories as groupoids for categories. On the other hand, the $\infty$...
Johnny's user avatar
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6 votes
1 answer
225 views

Mapping spaces in complete Segal spaces and quasi-categories

Complete Segal spaces and quasi-categories are two common models for the theory of $(\infty,1)$-categories, and both are equipped with a natural notion of hom spaces. For complete Segal spaces, which ...
ChrisLazda's user avatar
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5 votes
0 answers
177 views

Preservation of (co)limits under taking derived categories

Let $R$ be a commutative ring. Let $\{A_i\}_{i \in I}$ be a diagram of $R$-linear $1$-categories, indexed by a finite poset $I$. (If this matters, assume that the $A_i$ have finitely many objects). ...
Laurent Cote's user avatar
2 votes
1 answer
283 views

Is there a *relative* moduli stack of objects functor?

Toen and Vaquie have constructed for any dg category $\mathcal{C}$ a stack $\mathcal{M}_\mathcal{C}$ parametrising objects in $\mathcal{C}$. Its definition is $$\mathcal{M}_\mathcal{C}(R)\ =\ \text{...
Pulcinella's user avatar
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8 votes
1 answer
186 views

Sequential colimit of iterated quotients of Cauchy sequences

We work in constructive mathematics. The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
Madeleine Birchfield's user avatar
8 votes
0 answers
332 views

Has there been any progress on this open problem about co-well-poweredness of accessible categories?

On the relations between accessible categories and large cardinal axioms, one big example is the following: Assume the existence of a proper class of strongly compact cardinals. Then every accessible ...
interregno's user avatar
5 votes
0 answers
275 views

Serre subcategories of the category of chain complexes of modules

Let $k$ be an algebraically closed field of characteristic $0$. Let $R$ be a commutative $k$-algebra. We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...
Walterfield's user avatar
0 votes
1 answer
100 views

Categorical equivalences and essentially commutating squares

Let's take four categories $\mathcal{A},\mathcal{B},\mathcal{C}$, and \mathcal{D}. We will assume the existence of equivalences between them as laid out in the following diagram: Can it happen that ...
Quin Appleby's user avatar
3 votes
0 answers
113 views

Does symmetric product functor preserve fibrations?

I know that the symmetric product is a functor, cf: https://en.wikipedia.org/wiki/Symmetric_product_(topology)#Functioriality. My question is, does it preserve fibrations in the category of ...
JE2912's user avatar
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0 votes
1 answer
238 views

Trans-universality for finitely generated groups

QUESTION: does there exist a group U such that three conditions hold: (a) every finitely generated group is isomorphic to a subgroup of U; (b) for every group G that is not finitely generated there ...
Wlod AA's user avatar
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