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.
4,621
questions
1
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0answers
37 views
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 ...
2
votes
1answer
100 views
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.
...
9
votes
1answer
292 views
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 $...
2
votes
0answers
134 views
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 ...
5
votes
1answer
158 views
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 ...
4
votes
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:
$$...
6
votes
0answers
82 views
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 ...
2
votes
0answers
52 views
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 ...
3
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0answers
43 views
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....
3
votes
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)$ ...
6
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0answers
88 views
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 ...
1
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1answer
240 views
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 ...
6
votes
1answer
101 views
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 ...
5
votes
0answers
98 views
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) \...
3
votes
0answers
50 views
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 ...
4
votes
0answers
72 views
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 ...
9
votes
2answers
244 views
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 ...
2
votes
1answer
76 views
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-...
43
votes
4answers
4k views
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 ...
16
votes
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 ...
3
votes
1answer
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 $[X,B'G]$...
36
votes
1answer
1k views
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 ...
6
votes
1answer
158 views
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{...
6
votes
2answers
255 views
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 ...
-1
votes
3answers
124 views
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 ...
3
votes
0answers
94 views
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 ...
6
votes
3answers
374 views
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: $X \mapsto \pi_{\leq 1}(X)$, ...
5
votes
0answers
150 views
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 ...
1
vote
0answers
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 ...
7
votes
1answer
141 views
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 ...
5
votes
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 ...
3
votes
0answers
95 views
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 ...
8
votes
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 $...
2
votes
0answers
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 ...
4
votes
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:
$...
4
votes
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 ...
1
vote
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 \...
8
votes
0answers
116 views
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 ...
5
votes
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 ...
8
votes
2answers
621 views
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 ...
14
votes
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 ...
5
votes
0answers
205 views
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 ...
3
votes
1answer
116 views
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 ...
1
vote
0answers
114 views
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:
...
2
votes
2answers
227 views
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)...
4
votes
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 ...
2
votes
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 ...
3
votes
0answers
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"
...
9
votes
0answers
298 views
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 $...
4
votes
0answers
59 views
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 ...
