# 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.

5,588
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### Strictness of two operations on proarrow equipments

There are several equivalent definitions of a profunctor between categories $C$ and $D$. I'm interested in the following two:
A functor $C\times D^o \to \text{Set}$
A co-continuous functor between ...

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0
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### Well-behaved monad quotients

Reading through Modular specification of monads through higher-order presentations, this paper includes the following lemma within set-truncated homotopy type theory:
Given a monad $R$ (they work on ...

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3
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### Examples of five-adjoint systems

I'm looking into Lawvere's formulation of unities of opposites and opposites of unities, and for this I would be interested in systems of five (or more) adjoint functors $X\stackrel{\stackrel{\...

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### Set theory for category theory

Let $SC$ (sets and classes) denote the modification of Ackermann set theory laid out in F. A. Muller's Sets, Classes and Categories (denoted there by ARC -- a brief summary of the theory is given ...

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### Natural numbers object in a double category

I've been playing with double categories. I'm having trouble figuring out appropriate laws for induction squares in a double category.
Assume an object $\mathbb{N}$.
Assume horizontal arrows
zero $0\...

5
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1
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90
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### Is every folk cofibration of strict $\omega$-categories a monomorphism?

In the folk model structure on the category $sCat_\omega$ of strict $\omega$-categories, the cofibrations are generated by the boundary inclusions $\{\partial \mathbb G_n \to \mathbb G_n \mid n \in \...

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### Deformation of (locally) ringed spaces and of their abelian categories of modules

I am interested in the general theory of deformations locally ringed spaces in the same language of the deformation theory of schemes/varieties that is already widely available. I am aware for example ...

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### Strongly simple fusion categories: the known examples?

A fusion category is called simple if its fusion subcategories are just $Vec$ and itself. Let us call a fusion category strongly simple if every fusion category Morita equivalent to it (i.e. same ...

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### Naturally occurring examples of categories where composition depends on objects

In the comments and answer to another recent question, it became apparent that category theorists who work with the ‘many hom-class’ definition of a category implicitly view composition as a function ...

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### Examples and counterexamples to Lack's coherence observation

In Lack's A 2-categories companion, he states
There are general results asserting that any bicategory is biequivalent to
a 2-category, but in fact naturally occurring bicategories tend to be ...

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### Constructing new categories by adding structure

On the one hand, let $\mathcal{C}$ be the monoidal category of finite-dimensional complex vector spaces and linear transformations, and on the other, let $\mathcal{D}$ be the monoidal category of ...

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### Why does representing functors help solving Diophantine equations?

Here I read:
Another insight of Grothendieck and his school was, how important it is to represent functors in algebraic geometry - regardless of what you want at the end. [as Mazur reports, Hendrik ...

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### Does a well-pointed topos with enough projectives satisfy the internal axiom of chioice?

If yes, then I am also wondering if being well-pointed can be weakened to boolean (i.e. this is in the context of using Set as our metalogic so that well pointed Topoi are boolean). If not, then any ...

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2
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898
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### Missing axiom in the typed definition of a category?

$\newcommand\bHom{\mathbf{Hom}}\newcommand\bOb{\mathbf{Ob}}\newcommand\bRel{\mathbf{Rel}}$This question is probably stupid and definitely bureaucratic, but
Is writing $f\circ g$ for the composition ...

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144
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### A category with two notions of weak equivalence?

I have a category with two classes of weak equivalences (neither class is contained in the other, and I believe they actually both form homotopical categories). I'm wondering if this notion exists in ...

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### Grothendieck's relative point of view and Yoneda lemma

I asked this question on M.SE, but didn't get any answers.
Occasionally I hear people saying that one of Grothendieck's big insights was that often when interested in an object $X$ it's better to ...

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1
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### Is there a Dold-Kan theorem for circle actions?

There are several interesting equivalences of "Dold-Kan type" in the setting of stable $\infty$-categories. Namely, let $\mathcal C$ be a stable $\infty$-category. Then the following 3 ...

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### Is there a "duality involution" on presentable categories?

$\newcommand\Psh{\mathit{Psh}}\newcommand\Pr{\mathit{Pr}}$Let $\Psh$ be the category of presheaf categories and cocontinuous functors which preserve tiny objects. There is a functor $(-)^\ast : \Psh \...

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### Is $\mathit{Topos}^\text{op} \to \mathit{Pr}^L$ monadic?

$\newcommand\Logos{\mathit{Logos}}\newcommand\Topos{\mathit{Topos}}\newcommand\op{^\text{op}}\newcommand\Pr{\mathit{Pr}}$Let $\Logos = \Topos\op$ be the $\infty$-category of $\infty$-topoi and ...

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### Reference on subcategories of ${\bf Rel}$

Is there a reference discussing the 'canonical' subcategories of ${\bf Rel}$?
By 'canonical subcategories' I mean the categories $${\bf EntRel},$$ $${\bf FunRel},$$ $${\bf SurRel},$$ $${\bf InjRel},$...

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### Non-trivial automorphisms and descent

In this expository paper by Low it says:
Roughly
speaking, a topos in the sense of Grothendieck is the category of sheaves on a
kind of generalised space whose “points” may have non-trivial ...

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0
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### Transporting monoidal structure along adjunction

Let $L: \mathcal C \leftrightarrows \mathcal D : R$ be an adjoint pair and $\mathcal C = (\mathcal C,\otimes)$ be a monoidal category. I wonder under what conditions on the pair $L\dashv R$ we can ...

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### Is any $n$-angulated category a $(n-2)$-cluster tilting subcategory of some triangulated category?

Geiss, Keller and Oppermann told us in "n-angulated categories" that some $(n-2)$-cluster tilting subcategory of a triangulated category is a $n$-angulated category.
$\require{wasysym}$
...

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### Do systems of objects over a Grothendieck category form a Grothendieck category?

Let $\mathscr C$ be a Grothendieck category and let $I$ be a small category (not a preadditive category, just a small category). Is the category $\mathscr C^I$ of systems of objects in $\mathscr C$ ...

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### When forgetting structure doesn't matter

What forgetful functors are equivalences?
The motivation here is understanding when some part of a structure can be 'safely' forgotten, even if remembering it might make our lives easier.
There is ...

2
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0
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103
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### What is a closed monoidal metric space?

This recent MSE question started a conversation with the OP of that post about what are some categorical notions casted in the category of metric spaces, regarded as enriched categories over $[0,\...

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1
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102
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### Does left-exactness imply semi-additivity?

Let $\mathcal C$ and $\mathcal D$ be pre-additive (enriched over the abelian groups) categories and $F : \mathcal C \Rightarrow \mathcal D$ a functor which is left-exact in the sense that it preserves ...

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2
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### Existence of nontrivial categories in which every object is atomic

An object $X$ of a cartesian closed category $\mathbf C$ is atomic if $({-})^X \colon \mathbf C \to \mathbf C$ has a right adjoint (hence is also internally tiny). Intuitively, atomic objects are &...

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1
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### Linear logic and linearly distributive categories

I asked this question ten days ago on MathStackexchange (see here). Despite having placed a bounty on the question, I have not received any answers or comments until now. Following Nick Champion's ...

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### On the pro-category of finite local artinian algebras

Let $\mathbb{F}$ be a finite field, and $W(\mathbb{F})$ its associated ring of Witt's vectors. On page 6 of the following lecture notes Deformations of Galois Representations, the category $\mathfrak{...

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### Arithmetic theorems explainable to a high schooler using $\infty$-category theory in a crucial way

For some context, Fermat's last theorem has a beautiful statement, which can be explained to a high schooler, it appears naturally when teaching the pythagorean theorem and just a bit of ...

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1
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### What are the algebras for the laxification 2-monad?

Let $C$ be a small 2-category. Let $[C , Cat]$ denote the 2-category of strict functors to $Cat$, 2-natural transformations, and modifications. Let $[[ C, Cat ]]$ denote the 2-category with the same ...

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### Ultracategories with one object

Historically, the theory of ultracategories was invented by Makkai to prove a strong conceptual completeness theorem for first-order logic, roughly: if $T$ and $S$ are two first-order theories such ...

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286
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### How to show that $\text{Man}(M,\mathbb{R}^n)\cong \mathbb{R}\text{-Alg}(C^\infty(\mathbb{R}^n),C^\infty(M))$?

I'm trying to show that manifolds are affine, i.e. $\text{Man}(M,N)\cong \mathbb{R}\text{-Alg}(C^\infty(N),C^\infty(M)) $. If I could show this for $N=\mathbb{R}^n$, then I know how to do the rest ...

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### Category of entire relations

Is there a reference that talks about the category ${\bf EntRel}$ whose objects are sets and whose arrows are entire relations?
As a preliminary observation ${\bf EntRel}$ is complete and cocomplete, ...

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### EM functor from monads to adjunctions

What is the action on $1$-cells of the functor sending a monad to its EM adjunction? What about the Kleisli adjunction?
Let $A$ be the walking adjunction. Recall that an adjunction is the same thing ...

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2
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### Does there exist a complete algebraic invariant of the homotopy type of a finite CW-complex?

Let $\mathrm{Cell}$ be the homotopy category of finite cell complexes. The main motive of my question
Is it true that for any algebraic category $A$ there is no fully faithful functor $F: \mathrm{...

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### "Whenever we have some interesting invariant of spaces, we try to cook up a space that represents this invariant"

In his essay Classifying Spaces Made Easy
Baez writes:
We've seen this trick a couple of times lately, and it's actually a
big theme in homotopy theory: whenever we have some interesting
invariant of ...

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2
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709
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### Groupoid cardinality of the class of abelian p-groups

$\DeclareMathOperator\Aut{Aut}\newcommand\card[1]{\lvert#1\rvert}$So, after going over the classification of finite abelian groups in a class I was teaching this winter, I got curious about whether it ...

5
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1
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539
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### Adjunction between topological spaces and condensed sets

I am trying to prove that the functor
\begin{align*}
\mathrm{Top} &\longrightarrow \mathrm{Cond}(\mathrm{Set}) \\
X &\longmapsto \underline{X}
\end{align*}
admits a left adjoint and it is the ...

6
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0
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### Does the category of cosheaves have enough projectives?

Given a general topological space $X$ does the category $\mathbf{coShv}(X,\mathbf{Mod}_R)$ have enough projectives ? I know that under some conditions this is true, for example if $X$ is a cell ...

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### Infinite Galois equivalence

I am having some trouble trying to understand the proof of Theorem 7.2.5 in Bhatt and Scholze's paper The pro-étale topology for schemes. Specifically, I don't quite understand why it was necessary to ...

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### A recursive attempt at $n$-dimensional coherence

For the purposes of this post we will use the one hom class definition of a category.
Note that a functor $F:\mathcal{C}\to\mathcal{D}$ between categories is a pair of functions $F_0:{\bf Ob}_\mathcal{...

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1
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### Mathematical life of Friedrich Ulmer

In the little galaxy of Category Theory, Friedrich Ulmer is known for being one of the authors of Lokal Präsentierbare Kategorien, a book that laid the foundations for the theory of locally ...

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### Pullback and pseudoelements

Let $\mathcal{A}$ be an abelian category, and let $X$ an object of $\mathcal{A}$. Recall that a pseudoelement of $X$ is an equivalence class of arrows $X_1 \to X$, where $x_1 \colon X_1 \to X$ and $...

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### Can there be non-isomorphic fundamental groups of equivalent Galois categories?

It is known that if $(C, F)$ is a Galois category then there exists an equivalence $C \cong \pi_1(C, F)-FinSets$ between $C$ and the category of finite sets with continuous actions of $\pi_1(C, F) := ...

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### Summary of different types of TQFT?

For the purposes of this question, a TQFT comprises the following data:
An "upper dimenison" $n \in \mathbb N$.
A "lower dimension" $0 \leq l \leq n$.
A choice of structure ...

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### Lifting isomorphisms between linear categories

Let $C$ be a $\mathbb{Z}$-linear category, such that $C(x,y)$ is a free abelian group with finite rank, for every $x,y\in\mathrm
{Ob}(C)$. Given a commutative ring with identity $R$, let $RC$ denote ...

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### Group ring objects in a Cartesian closed category

Let $\mathcal{C}$ be a Cartesian closed category, with $R$ a ring object in $\mathcal{C}$ and $G$ a group object in $\mathcal{C}$.
Is there literature on the notion of the 'group ring object' $R^G$?
...

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### Homotopy fixed points vs coalgebras

Referring to the last part of this answer https://mathoverflow.net/a/225403/170683, I would like to understand how in the case of a Galois cover $f\colon X\to Y=X/G$ with Galois group $G$ (I guess ...