Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,366
questions
2
votes
0
answers
153
views
Equivariant objects of derived categories
Suppose $C$ is a $k$-linear abelian category with an action of a linear algebraic group $G/k$. Suppose $C$ has enough projectives/injectives so I can form the bounded derived category $D(C)$. Under ...
9
votes
2
answers
802
views
What norms can be "universally" defined on any real vector space with a fixed basis?
Let $V$ be a real vector space and let $B = (b_\lambda)_{\lambda \in \Lambda}$ be a basis. So every $v \in V$ can be written uniquely as a linear combination
$$ v = c_{\lambda_1} b_{\lambda_1} + c_{\...
7
votes
0
answers
278
views
Does the pentagon axiom force the associativity constraint to be a natural isomorphism?
Consider a fusion ring and the associated system of polynomial equations induced by the pentagon axiom of a fusion category. A solution of this system is supposed to encode the associativity ...
9
votes
2
answers
592
views
Abelian categories satisfying AB5*
I could name on the spot a bunch of abelian categories satisfying AB5 but I cannot think of any that satisfies AB5*. That is, it should have all limits and the cofiltered limits are exact. Is there ...
4
votes
2
answers
256
views
Is $\Delta^1 \times \Delta^1 \cup_{\partial \Delta^1 \times \Delta^1} \partial \Delta^1 \times I \to \Delta^1 \times I$ inner anodyne?
Let $I$ be the (nerve of the) walking isomorphism (a simplicial set). Consider the inclusion $\Delta^1 \to I$ and the inclusion $\partial \Delta^1 \to \Delta^1$. Take their pushout-product to obtain a ...
1
vote
0
answers
65
views
Does lambda polymorphism have some universal property?
To evaluate some typed lambda calculus applications, the type of the function might have to be "lifted" in order to match the type of the value it is applied to. For example, in the ...
11
votes
2
answers
785
views
Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property?
I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does ...
4
votes
1
answer
262
views
Mackey coset decomposition formula
I have a question about following argument I found
in these notes on Mackey functors:
(2.1) LEMMA. (page 6) Let $G$ be a finite group and $J$ any subgroup. Whenever $H$ and $K$ are subgroups of $J$, ...
1
vote
1
answer
91
views
Choosing a net of projections from a given collection
Let $\mathcal M$ be a von Neumann algebra. Let $S$ be a nonempty set. Consider $\Omega$ to be the collection of all finite subsets of $S$. Suppose for all $\omega\in\Omega$ we have $I_\omega$ a ...
5
votes
0
answers
216
views
Higher coherences in $A_\infty$-spaces
Recently I was learning about $A_\infty$-spaces. The slogan what they are is: "Homotopy associative H-spaces with higher coherences for the associativity".
When I first heard the slogan, I ...
5
votes
2
answers
400
views
Topos semantics of constructive higher order logic
I would like to find a reference that describes the semantics of constructive higher order logic with function types in toposes. In particular, it seems that if we are to take function types as ...
2
votes
0
answers
72
views
Are $\mathscr{V}$-modules uniquely (nicely) enrichable?
$\require{AMScd}\newcommand{\V}{\mathscr{V}}\newcommand{\M}{\mathcal{M}}\newcommand{\hom}{\operatorname{hom}}\newcommand{\op}{{^\mathsf{op}}}$Fix a closed symmetric monoidal category $(\V;\otimes;\...
0
votes
0
answers
101
views
Smoothness of a deformation functor
In his notes of deformation of complex structures Manetti defined deformation as a functor of Artin rings. He also defines the smoothness of this functor as
Definition V.36. Let $F, G: \mathbf{A r t}...
5
votes
0
answers
205
views
Lax monoidal structure on the right Kan extension of a partially monoidal Γ-set
First some preliminaries. Let me write $Fin_\ast$ for the skeleton of the category of finite pointed sets and pointed maps between them on the objects $n_+=\{0,1,...,n\}$, where $0$ is the base point (...
7
votes
1
answer
152
views
Is lambda calculus polymorphism a type of generalized monad?
Let $\mathbf{C}$ be a Cartesian closed category. Then simply typed lambda calculus in $\mathbf{C}$ in one type variable can be interpreted as a category $\mathbf{STLC}_{\mathbf C}$ where the objects ...
5
votes
0
answers
72
views
Braided monoidal category of (generalized) operator algebras
In the paper "Quantum Collections" Kornell (2016) proved that the category of $W^*$ (von Neumann) (and iirc also $C^*$) algebras, equipped with a suitable choice of tensor product, forms a ...
11
votes
1
answer
431
views
Recasting straightening/unstraightening equivalence as $(\infty, 2)$-adjunction
This is a vague set of questions that relies on (possibly non-existent) generalizations of low-dimensional results, mostly because I don't know many of the technical details underlying the ...
8
votes
0
answers
182
views
Ind-objects with "full support"
I originally posted this question here on Math Stackexchange, but I didn't get an answer.
Let $C$ be a small category. Let's say a presheaf $P\colon C^{\mathrm{op}}\to \mathsf{Set}$ has "full ...
7
votes
1
answer
275
views
structure in triangulated category similar to t-structure
It’s well known that the heart of a t-structure is an abelian category. My question is that can we find some structure on a triangulated category which can “produce” an exact category in analogy with ...
4
votes
1
answer
214
views
A cocomma square from the Kleisli category of a monad
$\require{AMScd}$Let $T : X\to X$ be a monad on a category $X$; define the category $X §_T X$ as follows: the objects are $X_0 + X_0'$, i.e. two disjoint copies of the objects of $X$, and the ...
7
votes
1
answer
252
views
Eilenberg-Moore category as a 2-dimensional limit
$\require{AMScd}$Given an endofunctor $F : C\to C$, its category of algebras is the inserter of $F$ and the identity functor. This means that there is a square
$$\begin{CD}
Alg(F) @>j>> C \\
...
12
votes
2
answers
800
views
An overview of mathematical-logical approaches in formalizing natural languages
Crossposted on Mathematics SE
I am an undergraduate mathematics student with a keen interest in pursuing research in the formalization of natural languages (from a more mathematical-logical approach),...
4
votes
0
answers
106
views
Equalizer-product formula for $(\infty, 1)$-limits
If $F : K \to C$ is a functor of ordinary categories and $C$ has products and equalizers, then there is an isomorphism
\begin{equation*}
\lim F \cong \mathrm{eq} \left( \prod_{k_0 \in K_0} F(k_0) \...
7
votes
0
answers
165
views
Who first introduced the term "categorical group", and when?
The term "categorical group" is often used to mean a group object in Cat; these days we also call such a thing a strict 2-group. Who first introduced the term "categorical group", ...
3
votes
1
answer
166
views
Enriched cofibrant replacement in spectrally enriched categories
If $\mathcal{V}$ is a monoidal model category with all objects cofibrant, Theorem 13.5.2. of Categorical Homotopy Theory will guarantee that the functorial cofibrant replacement of a $\mathcal{V}$-...
2
votes
1
answer
66
views
Nomenclature help: action vs. module and “pointed” monadic algebras are module objects?
I'm confused about some nomenclature in a paper by J. Goguen from 1975. I'm happy to use definitions however they appear. But I aim to summarise the paper for a non-expert audience and I want them to ...
0
votes
1
answer
181
views
Free enriched monoidal categories
Suppose $(\mathcal{V},\otimes,1)$ is a symmetric monoidal category and $\mathbb{C}$ is a $\mathcal{V}$-category. I will deliberately avoid usual powerful assumptions (eg completeness/cocompleteness) ...
4
votes
0
answers
86
views
Relationship between coarse objects, separated objects, and sheaves
I would like to better understand the relationship between quasitopoi and topoi. Here are two relationships that I am aware of:
Given a local topos $E \to S$, i.e. such that $S$ is equivalent to the ...
6
votes
0
answers
344
views
Can Langlands correpondence be restated using topos?
Langlands correspondence describes an equivalence between Galois representations and automorphic representations under some conditions.
Laurent Lafforgue applying Olivia Caramello thesis described in ...
4
votes
1
answer
141
views
Relationship between Kan extensions and internal hom
Let $\mathcal{C}$ be a (sufficiently complete and cocomplete) closed monoidal category with internal hom $[-,-]$. Let $F : \mathcal{A} \to \mathcal{C}$ be a functor obtained as the left Kan extension ...
7
votes
1
answer
262
views
Enriched 2-categories
I have been lead to believe, due to various conversations and presentations, that there is a standard notion of an enriched 2-category (indeed, even an enriched n-category). However, after searching I ...
0
votes
0
answers
68
views
Mating a morphism of pullbacks in $Cat$, when is this well-behaved?
In this cube of functors, the front and back faces are strict pullbacks
all the diagonal arrows are right adjoints: let's take the cube where the left adjoints have been considered, it is now filled ...
0
votes
0
answers
98
views
Understanding this (standard?) notion of enriched product category
$\newcommand{\V}{\mathscr{V}}\newcommand{\A}{\mathcal{A}}\newcommand{\B}{\mathcal{B}}\newcommand{\C}{\mathcal{C}}$Fix a closed symmetric monoidal category $\V$, writing the product as $\otimes$, the ...
9
votes
1
answer
493
views
Isbell Duality and Dualizing Scheme Objects
I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\...
2
votes
1
answer
158
views
Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?
Let $\mathcal{R}$ be a $1$-category. Assume that one has a pushout of representable $1$-presheaves $\mathrm{y} A \cup_{\mathrm{y} B} \mathrm{y} C$ in $\mathsf{PSh}(\mathcal{R})$. Under which ...
2
votes
0
answers
152
views
Can End(F) be viewed as a pro-object in the category of finite dimensional algebras?
In EGNO 1.10, we have essentially the following setup: given a $\mathbb{C}$-linear abelian category $\mathcal{A}$ and an exact faithful functor $F: \mathcal{A} \to Vec$ to the category of finite ...
7
votes
1
answer
137
views
Preservation of lax limits in categories of functors and lax natural transformations
Let $f:\mathbb{C} \to \mathbb{D}$ be a functor of 2-categories and let $\operatorname{Fun}(\mathbb{C},\operatorname{Cat})^{\operatorname{lax}}$ denote the 2-category of functors and lax natural ...
8
votes
2
answers
416
views
Condition for an equivalence of functor categories to imply an equivalence of categories
Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\...
6
votes
1
answer
427
views
Why isn't pullback-stability defined for individual colimits but for colimits with the same shape?
Circumstances: I'm studying Grothendieck's Galois Theory and recently encountered a proposition that discussed the stability of coproducts under pullback. And I found the page pullback-stable colimit ...
3
votes
0
answers
71
views
Refined Duskin Nerve
The Duskin nerve sends a 2-category C to the simplicial set whose $n$-th term
is the set of normal lax 2-functors $[n] \to C.$
The Duskin nerve gives a full embedding from the category of 2-categories ...
9
votes
3
answers
2k
views
Why is category theory the preferred language of advanced algebraic geometry?
It's pretty easy to see how category theory could be applied in the case of abstract algebra, but doing so doesn't seem particularly useful, at the very least, for purposes of the stuff that one finds ...
28
votes
5
answers
5k
views
What is the motivation for infinity category theory?
To my understanding, most mathematical theories can be simply understood in the view point of Category theory and its derivative theories. But what exactly is the motivation to study infinity category ...
2
votes
0
answers
68
views
Justification of modular law in allegories
The modular law in modular lattices can be described as an isomorphism between opposite edges of the square $(a \land b), a, b, (a\lor b)$. A fancier way of saying this is an adjoint equivalence with ...
5
votes
1
answer
2k
views
Do bijections from the natural numbers satisfy the Peano axioms? [closed]
While thinking of natural numbers as anything that satisfies the Peano axioms, I was left wondering, what if I take the successor function $S(x)$ to be anything other than $x\to x+1$?
Some examples ...
1
vote
1
answer
362
views
Limits of infinity categories and mapping spaces
Let $p:I\to Cat_{\infty}$ be a diagram of infinity categories, where $I$ is a small Kan complex. Let $C:=\lim p$ be the limit of $p$. For any two objects $x,y\in C$ and $i\in I$, let $x_i,y_i\in C_i=p(...
3
votes
0
answers
188
views
How to read the definition of Grothendieck Pretopology in SGA4?
In SGA4, the first axiom of a Grothendieck pretopology is given as:
PT0: Pour tout objet $X$ de $C$, les morphismes des familles de morphismes de $Cov(𝑋)$
sont quarrables. (Rappelons qu’un morphisme ...
3
votes
2
answers
103
views
Does unitarity and modularity constrain fusion multiplicities to be 0,1?
If I have a braided tensor category that's unitary and modular, then how does the unitarity and modularity constrain the fusion multiplicities?
I know that if $a,b,c \in ob({C})$ satisfy the fusion ...
4
votes
1
answer
193
views
Profinite groups with isomorphic proper, dense subgroups are isomorphic
I am developing a sort of standard representation for profinite quandles. This involves profinite groups a lot, actually. In one part of my construction the filtered diagram used to construct a ...
2
votes
0
answers
77
views
Free strict Picard groupoid
Suppose that $S$ is a set and that $\mathcal G(S)$ is a strict Picard groupoid endowed with a function $f\colon S \to \mathrm{ob}\mathcal G(S)$. For any other strict Picard groupoid $\mathcal G$, ...
3
votes
1
answer
256
views
Ultra*powers* in the category of structures and elementary embeddings
This is based on a few previous questions.
Can one characterize ultrapowers in the category of L-structures (modeling a fixed complete theory, say) and elementary embeddings?
Previous posts showed ...