Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,364
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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 \...
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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{...
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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 \...
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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 $...
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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 ...
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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$ ...
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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}$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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:\...
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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 ...
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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 ...
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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. ...
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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 ...
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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 ...
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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....
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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)....
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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: \...
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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 $p$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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960
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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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 ...
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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 $F : (\mathbb N,\le)^\text{op}\...
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Is there anyway to formulate the Alexandrov topology algebraically?
One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set.
Given this, one finds a one-to-one correspondence between ...
6
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Examples of bilimits that aren't 2-limits, and some related questions
Recently I've received an email from Sori Lee about an earlier question I had asked, and we ended up with a number of questions about 2-limits and 2-bilimits which I couldn't quite answer, and decided ...
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The category of groupoids vs the category of sets
Hopefully this question makes sense. As we know that Kan complexes are the "$\infty$-version" of groupoids for $\infty$-categories as groupoids for categories. On the other hand, the $\infty$...
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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 ...
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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).
...
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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{...
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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. ...
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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 ...
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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 ...
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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 ...
3
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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 ...
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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 ...