Questions tagged [ct.category-theory]

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

Filter by
Sorted by
Tagged with
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 ...
user333154's user avatar
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_{\...
AlpinistKitten's user avatar
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 ...
Sebastien Palcoux's user avatar
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 ...
user141099's user avatar
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 ...
Tim Campion's user avatar
  • 60.6k
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 ...
Johan Thiborg-Ericson's user avatar
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 ...
Tian Vlašić's user avatar
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 $J$, ...
user267839's user avatar
  • 5,948
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 ...
A beginner mathmatician's user avatar
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 ...
AlexE's user avatar
  • 2,926
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 ...
Trebor's user avatar
  • 1,021
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;\...
FShrike's user avatar
  • 569
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}...
user24918's user avatar
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 (...
Jonathan Beardsley's user avatar
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 ...
Johan Thiborg-Ericson's user avatar
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 ...
xuq01's user avatar
  • 1,054
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 ...
Lorenzo Riva's user avatar
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 ...
Alex Kruckman's user avatar
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 ...
Yifei Cheng's user avatar
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 ...
fosco's user avatar
  • 13k
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 \\ ...
fosco's user avatar
  • 13k
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),...
Heleyrine Brookvinth's user avatar
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) \...
Lorenzo Riva's user avatar
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", ...
John Baez's user avatar
  • 21.3k
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 $\mathcal{V}$-...
Connor Malin's user avatar
  • 5,191
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 ...
Timtro's user avatar
  • 123
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) ...
Morgan Rogers's user avatar
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 ...
Jonathan Sterling's user avatar
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 ...
jaylooker's user avatar
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 ...
Lorenzo Riva's user avatar
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 ...
Arthur's user avatar
  • 1,379
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 ...
fosco's user avatar
  • 13k
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 ...
FShrike's user avatar
  • 569
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}\...
LiminalSpace's user avatar
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 ...
HDB's user avatar
  • 355
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 ...
Chris's user avatar
  • 264
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 ...
Abellan's user avatar
  • 295
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 $\...
Cameron's user avatar
  • 81
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 ...
Linuxmetel's user avatar
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 ...
willie's user avatar
  • 499
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 ...
tryst with freedom's user avatar
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 ...
tryst with freedom's user avatar
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 ...
Trebor's user avatar
  • 1,021
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 ...
Povilas's user avatar
  • 129
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(...
Kim's user avatar
  • 505
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 ...
Joey Eremondi's user avatar
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 ...
pyroscepter's user avatar
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 ...
Alex Byard's user avatar
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 $\mathcal G$, ...
Alexander Betts's user avatar
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 ...
Pteromys's user avatar
  • 151

1
4 5
6
7 8
128