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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.

3
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0answers
45 views

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
1answer
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
1answer
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
1answer
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
1answer
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 ...
7
votes
1answer
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 $\...
10
votes
3answers
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 ...
3
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0answers
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
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5answers
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
votes
0answers
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 ...
4
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0answers
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 (...
11
votes
3answers
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 ...
5
votes
0answers
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
1answer
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 ...
2
votes
0answers
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 ...
8
votes
4answers
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 ...
16
votes
1answer
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
0answers
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
0answers
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
1answer
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 (...
6
votes
2answers
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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0answers
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 ...
1
vote
0answers
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 $\...
4
votes
0answers
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 ...
4
votes
0answers
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 ...
1
vote
1answer
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
1answer
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....
1
vote
1answer
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 ...
8
votes
3answers
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 ...
6
votes
3answers
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
0answers
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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votes
0answers
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 ...
7
votes
2answers
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)\...
4
votes
0answers
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 ...
5
votes
0answers
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 $$ \Omega^{d}_{H}(BG)= \mathbb{Z}_m \oplus \dots, $$ ...
8
votes
2answers
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 ...
6
votes
3answers
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{...
11
votes
1answer
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 ...
1
vote
0answers
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 ...
2
votes
1answer
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 ...
6
votes
1answer
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....
4
votes
0answers
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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votes
0answers
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 $f: X \rightarrow Y$ ...
8
votes
1answer
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 ...
7
votes
0answers
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 ...
11
votes
1answer
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
1answer
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
0answers
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 $j$.
3
votes
1answer
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 ...
2
votes
1answer
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 $(...