# 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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### filetered colimit of fibrant-cofibrant objects

Suppose that we have a $\lambda$-combinatorial model category $M$ (for some cardinal $\lambda$) such that any $\lambda$-filtered colimit of fibrant-cofibrant objects is always fibrant. My question is ...

**5**

votes

**1**answer

237 views

### Universal property of sheaf category

Given a site $C$ with a Grothendieck topology and the category of presheaves $P(C)$ (either in the sense of presheaves of sets or in the $\infty$-sense), and the category $S(C)$ of sheaves with ...

**8**

votes

**1**answer

218 views

### Functoriality of (co)limits in $\infty$-categories

I have some questions about the functoriality of (co)limits in $\infty$-categories, say in the framework of Lurie's Higher Topos Theory.
From the general stuff about Kan-extensions (HTT 4.3.2.6) ...

**5**

votes

**1**answer

173 views

### Comonad for normalized pseudofunctors for strict higher categories

Garner constructed (in [1] ) for the category of strict $n$-categories a comonad $Q$ as the left part of a cofibrantly generated algebraic weak factorization system such that $$n\text{-}\operatorname{...

**2**

votes

**1**answer

75 views

### Definition of $\in_c$ for power objects

On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any ...

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**1**answer

266 views

### Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?

Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...

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votes

**3**answers

681 views

### Are inclusions “canonical” injections?

[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question]
Summary of question: the inclusions are a particularly ...

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**0**answers

49 views

### Do we have criteria of strict localization of a Grothendieck category?

Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...

**21**

votes

**5**answers

1k views

### are quotients by equivalence relations “better” than surjections?

This might be a load of old nonsense.
I have always had it in my head that if $f:X\to Y$ is an injection, then $f$ has some sort of "canonical factorization" as a bijection $X\to f(X)$ followed by an ...

**3**

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**0**answers

81 views

### How to show mapping cones are homotopy cofibers

In a dg-category $\mathcal{C}$, the $n$-translation of an object $C$ is an object $C[n]$ representing the functor
$$
{\rm Hom}(-,C)[n].
$$
The cone of a closed morphism $f\colon C \to D$ of degree ...

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**0**answers

151 views

### How to construct the espace étalé (space of sections) for an arbitrary category?

I want to consider the sheaf valued in an arbitrary category (not only of sets, groups, modules and so on) on a topological space, using the language of étalé space.
In all references I am reading (...

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**3**answers

631 views

### Comparisons of convenient categories for algebraic topology

I've heard that there are many convenient categories for algebraic topology. Such categories often have many nice properties like being cartesian closed, complete, cocomplete, the forgetful functor ...

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**0**answers

135 views

### Dugger and Spivak's combinatorial proof of Joyal's isofibration theorem, a fortiori?

In what follows, let $E^n = \operatorname{cosk}_0(\Delta^n)$
Joyal's isofibration theorem says precisely
An inner fibration $p:X\to Y$ of quasi-categories is a fibration for the Joyal model ...

**12**

votes

**1**answer

223 views

### Tilting Objects in BGG Categories $\mathcal{O}$

Let $\mathcal{O}$ be the BGG category $\mathcal{O}$ with respect to a finite dimensional semisimple Lie algebra $\mathfrak{g}$ and its Borel subalgebra $\mathfrak{b}$ (as define in this book by ...

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**0**answers

72 views

### Some notions of immersions of locally ringed spaces

Let $f:X\to Y$ be a morphism of locally ringed spaces. In this MSE answer, the first definition below is suggested.
Say $f:X\to Y$ is an $R$-immersion of locally ringed spaces if it's a topological ...

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votes

**4**answers

375 views

### Upgrade adjunction to equivalence

I'm studying category theory by myself and I just came across this sentence from Wikipedia:
An adjunction between categories C and D is somewhat akin to a "weak form" of an equivalence between C and ...

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votes

**1**answer

305 views

### Equivalences of categories of sheaves vs categories of $\infty$-Stack

Let say I have two different sites $(\mathcal{C},I)$ and $(\mathcal{D},J)$ for an ordinary topos $\mathcal{T}$. I.e.
$$Sh(\mathcal{C},I) \simeq \mathcal{T} \simeq Sh(\mathcal{D},J)$$
And we want to ...

**5**

votes

**0**answers

106 views

### Topos with enough projectives

It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...

**2**

votes

**0**answers

138 views

### Diagonal is representable then composition is representable

Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...

**3**

votes

**1**answer

237 views

### Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that,
If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
I could not find the statement (...

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votes

**2**answers

577 views

### Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....

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votes

**0**answers

126 views

### Statistical models of functions

I did a quick literature search and found nothing on "statistical models of functions". Let me explain what I am looking for. Given the category of Sets and Function, we have arbitrary functions ...

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vote

**0**answers

91 views

### Image of morphism of quasi-categories

I have two questions about images of morphisms of quasi-categories.
Suppose that $f\colon X \to Y$ is a morphism of quasi-categories.
Suppose that we calculate the image of $f$ in the category $\...

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votes

**0**answers

221 views

### Quillen pair, fibrant-cofibrant objects

This question is a follow-up of the question I have asked today : left quillen functor and fibrant objects
Suppose that we have $$ L :C\leftrightarrow D: R$$
an adjoint Quillen pair. We assume that ...

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votes

**0**answers

97 views

### Formalization and set-theoretic issues in the definition a functor category

Universes are used in a category theory to handle size-issues. Assuming Grothendieck's axiom UA that
every sets is contained in some universe
there are two approaches to $U$-smallness given a ...

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vote

**1**answer

106 views

### left quillen functor and fibrant objects

Suppose that we have $$ L :C\leftrightarrow D: R$$
an adjoint Quillen pair. We assume that both model categories are combinatorial model categories.
Suppose that the functor $L$ (left adjoint) takes ...

**9**

votes

**1**answer

355 views

### Is the Thomason model structure on Cat simplicial? Is it a monoidal model category?

The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e....

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vote

**1**answer

97 views

### Tensoring with complex of finite flat dimension in derived category

Let $(R,m)$ be a Noetherian local ring, and $X$, $Y$ be complexes of finitely generated $R$ modules. Suppose $X$ is bounded above and $Y$ is bounded below. Let $S$ be an $R$-algebra of finite flat ...

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**3**answers

211 views

### $\mathscr{U}$-categories and $\mathsf{Hom}$-functors

Let $\mathscr{U}$ be a universe. Call a set $X$ $\mathscr{U}$-small if there is a set $Y \in \mathscr{U}$ so that $X \cong Y$. Call a category $\mathsf{C}$ a $\mathscr{U}$-category if for any $X,Y \in ...

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votes

**3**answers

291 views

### Can natural section/retraction be checked pointwise?

Analogously to
this old question,
I was asking myself if it is possible to describe left/right invertible natural transformations by their components. Obviously this property is inherited by the ...

**3**

votes

**0**answers

102 views

### is the functor of finite order elements in grp representable

My question is whether the functor F:Grp->Set that sends a Group to its Set of elements with finite order is representable.
I have the sense that it shouldn't be but I've so far failed to prove it in ...

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108 views

### The category of multisets and spans

I have been thinking about multisets for a while. These are sets where elements can repeat, so $S =\{ a,a,b,c,b\}$ is a multiset on the set $A = \{a,b,c\}$.
I have also been looking into morphisms ...

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votes

**2**answers

206 views

### “Closed bicategories”

I am interested in the following property that a bicategory may or may not have.
Let $\mathbf{B}$ be a bicategory. Every one-morphism $f\colon x\rightarrow y$ defines a functor $\mathbf{B}(y,z)\...

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votes

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53 views

### Is there an analogue of final functors for genuine 2-categorical limits

A functor $I\to J$ of $1$-categories is called final, if each undercategory $(j, I)$ is connected.
More generally, for $(\infty,1)$-categories there is an analogous notion where one requires the ...

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124 views

### Categorification-like statement in the cobordism group?

Suppose we consider a $d$-cobordism group classifying manifolds with $H$-structure and with a classifying space $BG$ of a group $G$, written as
$$
\Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots,
$$
...

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votes

**2**answers

336 views

### Can one recover an algebraically closed field $k$ from the dots and arrows of its category of finitely generated $k$-algebras?

You are gracious enough to host me for a few days while I attend a conference. After I leave, you're surprised to see a gift on the kitchen table. It's a box with a category inside! The objects aren't ...

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**3**answers

338 views

### How do we formally construct the successor universe $\mathscr{U}^+$ of a universe $\mathscr{U}$ in $\mathsf{ZFC}$?

A set $\mathscr{U}$ is a universe if the following conditions are met:
For any $x \in \mathscr{U}$ we have $x \subseteq \mathscr{U}$
For any $x,y \in \mathscr{U}$ we have $\{x,y\} \in \mathscr{...

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votes

**1**answer

354 views

### Translating Grothendieck axiom UB into ZFC

In SGA 4, Grothendieck introduced a set-theoretic device called a Grothendieck universe. He and his collaborators worked in Bourbaki set theory, which is practically similar to $\mathsf{ZFC}$ but ...

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**0**answers

54 views

### Descent data and trivialization of bundles via coherent isomorphisms of fibers

In this MO question I tried to understand how a trivialization of a bundle (continuous map) $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$ is related to a coherent family of isomorphisms ...

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votes

**1**answer

220 views

### Limits, colimits and universes

For many purposes in category theory, we consider limit and colimits of diagrams $F\colon\mathsf{J\to C}$ where $\mathsf{J}$ is small category, that is, a category where the classes of objects and ...

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votes

**1**answer

123 views

### Derivator of localizations of spectra

Since my question is very specific let me introduce the context. I am trying to apply derivator theory to stable homotopy theory.
Theorem 3 of the following preprint by Franke
https://pdfs....

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**0**answers

89 views

### Free symmetric monoidal category of compactly generated category is compactly generated

Let $k$ be a field and let $\mathcal{C}=\mathbf{StLin}_k$ be the $\infty$-category of stable infinity categories enriched over the $\infty$-category $\mathbf{Vect}_k$, regarded as a symmetric monoidal ...

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27 views

### Gaussian Processes and the Measures of Finite Support Monad

Gaussian processes are defined on a domain, which is a set, and map each set element to a gaussian distribution over a target domain. I believe the idea is to model a function $f: X \rightarrow Y$ ...

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votes

**1**answer

169 views

### Is there a version of the “infinitary” disjunctive normal form theorem for topoi and slice categories?

According to nLab
Proposition 2.3. A complete Boolean algebra is completely distributive iff it is atomic (a CABA), i.e., is a power set as a Boolean algebra.
This is basically an infinitary ...

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votes

**0**answers

58 views

### Is the double orthogonal of a localizing Serre subcategory still a localizing Serre subcategory?

Let $R$ be an unital ring and consider the category of left $R$-modules $R$-Mod. We call a full subcategory $\mathcal{W}\subset R$-Mod a localizing Serre subcategory if $\mathcal{W}$ is closed under ...

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**1**answer

239 views

### “Scott completion” of dcpo

If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U $ for ...

**7**

votes

**1**answer

248 views

### The universal property of composition of morphisms

$\def\K{\mathcal K}$
Preamble.
Given a locally small category $\mathcal K$, its "composition law" is a class of maps
$$
c_{abc} : \K(a,b)\times\K(b,c)\to \K(a,c)
$$
with the universal property of an ...

**4**

votes

**0**answers

85 views

### Algebras for the codensity monad

Given a functor $j : A \to B$, when this extension exists we call $T_j = Ran_jj$ the codensity monad of $j$.
I was wondering if there's a general rule to guess the shape of $Alg(T_j)$ given $j$.

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**1**answer

174 views

### Locally presentable categories

Under category
Let $C$ be a locally presentable category, and let $c$ be an object of $C$. Lets denote by $C^{/c}$ the under category, objects are maps $c\rightarrow x$ and morphisms are the evident ...

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votes

**1**answer

158 views

### Localization of a model category with respect to a class of maps

I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:M\leftrightarrow N: R $$
and let $(...