Questions tagged [ct.category-theory]

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

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Consideration of large and small categories in Algebraic/differentiable/topological stacks

Angelo Vistoli in the notes Notes on Grothendieck topologies, fibered categories and descent theory starts the section of category theory with the following note: We will not distinguish between ...
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1answer
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What does play the role of a subobject classifier for quotient objects?

It is known that in the category of sets the dualization of the notion of a subobject classifiers does not work because the only object admitting a morphism into an initial object is the empty set. ...
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Are compact objects in presheaf categories finite colimits of representables?

An object $x$ in a category $\mathsf{C}$ is called compact or finitely presentable if $$\mathrm{hom}(x,-) : \mathsf{C} \to \mathsf{Set}$$ preserves filtered colimits. This concept behaves best when $...
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Descent and co/ends

The bicategorical analogue of a coend, namely a universal extrapseudonatural transformation, is called a bicodescent object. As noted in arXiv:1709.01332 [math.CT], this notion goes back to Ross ...
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1answer
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Applications of model categories

I was wondering if someone could explain some of the concrete applications of model categories. My possibly naive understanding of the motivation is that one wants to mimic the category of ...
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1answer
100 views

Categorical Kähler differentials and the Leibniz rule

From nlab, the module of Kähler differentials over some category $\mathcal{C}$ is the free functor: $$\Omega: \mathcal{C} \to \mathsf{Mod_{\mathcal{C}}}$$ left-adjoint to the (forgetful) embedding: $$...
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Any comparison between the category of cubes and its opposite?

To model topological spaces combinatorially, one can use simplicial sets -- or cubical sets. Simplicial sets are defined as presheaves on the simplex category $\Delta$, the category of non-empty ...
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A morphism of monads that doesn't preserve thunkability?

Recall that for a monad $(T,\eta,\mu)$ on a category $C$, the Kleisli category $C_T$ has as objects the objects of $C$ and as morphisms $C_T(x,y) = C(x,T y)$. A morphism $f\in C_T(x,y) = C(x,T y)$ is ...
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Locating the typed version of Hoàng Xuân Sính's thesis on Gr-categories

Hoàng Xuân Sính was a vietnamese student of Grothendieck who defended her thesis on Gr-categories (now called weak 2-groups). The thesis, handwritten and in French, can be found at https://pnp....
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1answer
108 views

Definition of the Yoneda Ext

Let $\mathcal{A}$ be an abelian category and let $X$ and $Y$ be objects in $\mathcal{A}$. The Yoneda $\text{Ext}^{n}(Y,X)$ is defined by the following: First we consider the class $\text{E}^{n}(Y,X)$ ...
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Examples of connection preserving maps in differential geometry

In synthetic differential geometry and tangent categories, linear connections on the tangent bundle are treated as a sort of algebraic gadgets that incorporate the tangent bundle. Like any other ...
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What is a good definition of a mathematical structure?

At the moment I am writing a textbook in Foundations of Mathematics for students and trying to give a precise definition of a mathematical structure, which is the principal notion of structuralist ...
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Notions of Lie 2-groupoids

The term Lie $2$-groupoid is used in the literature in more than one context. Some examples are given below: Ginot and Stiénon's paper $G$-gerbes, principal $2$-group bundles and characteristic ...
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Is the module of Kähler differentials a coend?

Let $\phi\colon R\to S$ be a ring map. The module of Kähler differentials $\Omega_{S/R}$ of $\phi$ can be constructed as the following coequaliser: $$\left(\bigoplus_{(a, b)\in S^2} S[(a, b)]\right) \...
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Applications of Day convolution for monoidal bicategories

Day convolution is a very powerful tool to build monoidal structures on categories of functors from a pro/monoidal $\mathcal{V}$-category. For instance, it is used in stable homotopy theory to ...
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2-categorical constructions preserving/inducing Yoneda structures

I'm curious about how Yoneda structures on a 2-category $\mathcal{K}$ play with various 2-categorical constructions. For example, if I have a 2-(co)monad on $\mathcal{K}$, are there any conditions ...
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The relation between t-structures and derived category

Let $\mathcal{D}$ be a triangulated category and a $t$-structure $(\mathcal{D}^{\leq 0},\mathcal{D}^{\geq 0})$ on $\mathcal{D}$. The heart of the $t$-structure, $\mathcal{A}=\mathcal{D}^{\leq 0} \cap ...
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1answer
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When uniquely divisible objects can be embedded into ind-torsion ones?

Let $A$ be an AB3 abelian category. We will say that an object $M$ of $A$ is uniquely divisible if for any integer $n\neq 0$ the endomorphism $nid_M$ is invertible. We will say that $M'$ is ind-...
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Category theory and set theory: just a different language, or different foundation of mathematics?

This is a question to research mathematicians, as well as to those concerned with the history and philosophy of mathematics. I am asking for a reference. In order to make the reference request as ...
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1answer
291 views

Topos extensions

In set theory, starting from a model $V$ of $ZFC$, a forcing notion $\mathbb{P}$, and a generic filter $G \subset \mathbb{P}$ over $V$, we can find a generic extension which is a model of $ZFC$ and is ...
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163 views

What can be an appropriate notion of principal bundle over a category (with an appropriate notion of local trivialisation)?

Motivation for my question: It is a well-known fact that there exists a bijection between the set of isomorphism class of principal $G$ bundles over a nice topological space $X$ and the set $[X,B'G]$...
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In the category of sigma algebras, are all epimorphisms surjective?

Consider the category of abstract $\sigma$-algebras ${\mathcal B} = (0, 1, \vee, \wedge, \bigvee_{n=1}^\infty, \bigwedge_{n=1}^\infty, \overline{\cdot})$ (Boolean algebras in which all countable joins ...
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1answer
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Universal model category as a $\text{sSet}$-enriched co-completion

It's a standard fact that given a small category $\mathcal{C},$ the category of pre-sheaves $\text{Psh}(\mathcal{C})$ is the free co-completion of it. I'm sure this can be done not only for $\text{...
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2answers
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Does the category of local rings with residue field $F$ have an initial object?

Let $F$ be a field. Does the category $C_F$ of local rings $R$ equipped with a surjective morphism $R\longrightarrow F$ have an initial object? This is, for instance, true if $F=\mathbb{F}_{p}$ for ...
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3answers
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Injective maps and direct limits [closed]

I have the following question. Assume you have a category $\mathcal{C}$ such that direct limits exists. Let $(C_n)_{n\in\mathbb{N}}$ be a sequence of objects in $\mathcal{C}$ and consider the ...
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Can chain homotopy induce space homotopy at $E_\infty$ level?

Space-level homotopy induces (co)chain homotopy, but I've never heard of the converse. I am not sure if it is simply not true? However, for good enough spaces (finite type nilpotent), Mandell proved ...
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3answers
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What is the geometric realization of the the nerve of a fundamental groupoid of a space?

It can be easily seen that there exists a functor $F:Top \rightarrow Grpd$ from the category of topological spaces to the category of groupoids defined as follows: Obj: $X \mapsto \pi_{\leq 1}(X)$, ...
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The $\infty$-category of natural transformations as an end

Let $\mathcal{C}$ be an $\infty$-category viewed as a fibrant scaled simplicial set with all 2-simplices thin and let $\mathfrak{C}\!at_{\infty}$ be the $\infty$-bicategory of $\infty$-categories. A ...
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120 views

Symmetric monoidal structure of the heart of $S^1$-spectra

How to give a symmetric monoidal structure of $SH^{S^1}(k)^{\heartsuit}$ (after $\mathbb{A}^1$-localization)? The standard answer is $$E_1\otimes E_2:=(E_1\wedge E_2)_{[0,0]}$$ but I don't see why ...
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1answer
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Categories of modules generated under coproducts by a small set?

Question 1: For which rings $R$ does there exist a small set $S \subseteq Mod_R$ such that every module $M \in Mod_R$ is a direct sum of modules in $S$? Equivalenty, for which rings $R$ does there ...
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1answer
208 views

Any exact faithful functor is represented by a unique projective generator

In the book 'Tensor Categories' by Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik on page 10 it says: 'Conversely, it is well known (and easy to show) that any exact faithful functor ...
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Is there a fusion category not Grothendieck equivalent to a unitary one?

We refer to the book Tensor categories by Etingof-Gelaki-Nikshych-Ostrik (MR3242743) for the notion of (unitary) fusion category. Two fusion categories are Grothendieck equivalent if they have the ...
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1answer
198 views

Free category with product and coproduct

Is there a known description of the free category with both product and coproduct ? That is given a small category $C$, I want to consider a category $U C$ which has product and coproduct, a functor $...
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80 views

Rings whose finitely-generated modules are co-hopfian

Let $A$ be a unital, possibly noncommutative ring. Dischinger showed [1] that the following are equivalent: For every $a \in A$, there exists $n \in \mathbb N$ such that $a^n A = a^{n+1} A$; For ...
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1answer
289 views

What is the notion of a group object and its action in a 2-category?

It is well known that a group object in a category $C$ (with terminal object $1$ and such that any two objects of $C$ have a product) is defined as an object $G$ in $C$ with the following morphisms: $...
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1answer
124 views

Is the center of an abelian rigid monoidal category, abelian?

Is the Drinfeld-Majid center of an abelian rigid monoidal category, abelian? [stated in 1J of On the center of fusion categories" by Bruguières and Virelizier (link at Virelizier's page)] In ...
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1answer
219 views

injectivity of pushout?

We have the following pushout diagram: $$\begin{array}{ccc} \langle X, Y \rangle & \xrightarrow{\alpha} & \mathbb{Z}_a \ast \mathbb{Z}_b \ast \mathbb{Z}_c \ast \mathbb{Z}_d \\ \downarrow \...
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Does forgetting colimits preserve colimits?

For each regular cardinal $\kappa$ let $\operatorname{Cat}_{\kappa}$ be the $(2,1)$-category of small categories with $\kappa$-small colimits, and functors that preserve those colimits. For each pair ...
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1answer
113 views

Adjunctions capturing duality between ideals and saturated monoids in a commutative ring?

Let $R$ be a commutative ring. A saturated monoid in $R$ is a multiplicative submonoid $S\subset R$ which is closed under divisors, i.e $xy\in S\implies x\in S$. This is the converse of the analogous ...
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Is the tensor product of chain complexes a Day convolution?

Recently, Jade Master asked whether the tensor product of chain complexes could be viewed as a special case of Day convolution. Noting that chain complexes may be viewed as $\mathsf{Ab}$-functors from ...
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1answer
690 views

What is a module over a Boolean ring?

Recall that a (unital) Boolean ring is a (unital) commutative ring $A$ where every element is idempotent; it follows that $A$ is of characteristic 2. There is an equivalence of categories between ...
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Kan extensions between categories of monoid objects

Let $K\colon\mathcal{A}\longrightarrow\mathcal{B}$ be a functor between $\mathcal{V}$-enriched categories. Endowing $\mathcal{A}$ and $\mathcal{B}$ with promonoidal structures, we obtain induced ...
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1answer
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Literature on linear categories

I am trying to understand Deligne's 'Categories Tensorielles', and therefore I need some knowledge on linear categories. Looking at Wikipedia and nLab, I found some definitions and explanations, but I ...
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Confusion in understanding the notion of $G$ Principal bundle where $G$ is a geometric group over a site

The first paragraph of the section Overview in the paper Principal infinity-bundles - General theory by Nikolaus, Schreiber and Stevenson https://arxiv.org/abs/1207.0248 precisely reads the following: ...
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2answers
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What is the geometric description of the set of isomorphism class of $G$-torsors over a site $C$?

Let $X$ be a topological space and $G$ be a topological group. Let $\tilde{G}$ be the sheaf of groups defined by the sheaf of sections of the product $G$ bundle $ \pi_1:X\times G \rightarrow X $. (1)...
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1answer
123 views

Projective (or injective) object in a subcategory

Let $\mathcal{A}$ be an abelian category and $\mathcal{B}$ be a full subcategory of $\mathcal{A}$. Suppose that $\mathcal{B}$ is abelian and that the inclusion of $\mathcal{B}$ in $\mathcal{A}$ is ...
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1answer
142 views

Representaility of morphism of stacks for schemes

I have seen two definitions of representability of a morphism of stacks, which should be at least compatible with the definition of a morphism of categories fibered in groupoids. (Representable ...
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45 views

Are the tangle functors based off Khovanov homology braided monoidal functors?

I was wondering if the tangle functors constructed in "A functor-valued invariant of tangles" https://arxiv.org/pdf/math/0103190.pdf "An invariant of tangle cobordisms via subquotients of arc rings" ...
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Is the morphism of sheaves $(R \mapsto GL(R((h)))) \rightarrow (R \mapsto PGL(R((h))))$ surjective in Zariski topology?

Consider two functors given by $R \mapsto GL(R((h)))$ and $R \mapsto PGL(R((h)))$ for a ring $R$. It is easy to see that these functors are sheaves in Zariski topology (in fact for any affine variety $...
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Need to know if a certain full subcategory of Top is cartesian closed

Consider the full subcategory of Top consisting of all spaces $X$ such that a subset $A$ of $X$ is closed if and only if $A \cap K$ is closed in $K$ for all subspaces $K$ of $X$ which are countably ...

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