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
votes
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 $$n\text{-}\operatorname{...
2
votes
1answer
75 views
Definition of $\in_c$ for power objects
On the nLab page for power objects, the object $\in_c$ is defined as the domain of a monomorphism $\in_c\hookrightarrow c\times\Omega^c$, and it is mentioned at the end of the article that in any ...
7
votes
1answer
266 views
Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?
Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...
10
votes
3answers
681 views
Are inclusions “canonical” injections?
[Background: I asked a vague question the other day, but as a result of the answers, particularly Andrej Bauer's, I now have a precise question]
Summary of question: the inclusions are a particularly ...
3
votes
0answers
49 views
Do we have criteria of strict localization of a Grothendieck category?
Let $\mathcal{C}$ be an abelian category and $\mathcal{S}$ be a full subcategory of $\mathcal{C}$. We call $\mathcal{S}$ a Serre subcategory of $\mathcal{C}$ if $\mathcal{S}$ is closed under ...
21
votes
5answers
1k views
are quotients by equivalence relations “better” than surjections?
This might be a load of old nonsense.
I have always had it in my head that if $f:X\to Y$ is an injection, then $f$ has some sort of "canonical factorization" as a bijection $X\to f(X)$ followed by an ...
3
votes
0answers
81 views
How to show mapping cones are homotopy cofibers
In a dg-category $\mathcal{C}$, the $n$-translation of an object $C$ is an object $C[n]$ representing the functor
$$
{\rm Hom}(-,C)[n].
$$
The cone of a closed morphism $f\colon C \to D$ of degree ...
4
votes
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)$$
And we want to ...
5
votes
0answers
106 views
Topos with enough projectives
It is often observed that every presheaf topos has enough projectives, as a corollary of the result that representables are projective and every presheaf is a colimit of representables. We also have ...
2
votes
0answers
138 views
Diagonal is representable then composition is representable
Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...
3
votes
1answer
237 views
Diagonal is representable then any morphism is representable
Ariyan Javanpeykar said here in comments that,
If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
I could not find the statement (...
6
votes
2answers
577 views
Theorem 2.1.2.2 Higher Topos Theory
At the page 74 of HTT, there is the following theorem
Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
0
votes
0answers
126 views
Statistical models of functions
I did a quick literature search and found nothing on "statistical models of functions". Let me explain what I am looking for. Given the category of Sets and Function, we have arbitrary functions ...
1
vote
0answers
91 views
Image of morphism of quasi-categories
I have two questions about images of morphisms of quasi-categories.
Suppose that $f\colon X \to Y$ is a morphism of quasi-categories.
Suppose that we calculate the image of $f$ in the category $\...
4
votes
0answers
221 views
Quillen pair, fibrant-cofibrant objects
This question is a follow-up of the question I have asked today : left quillen functor and fibrant objects
Suppose that we have $$ 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 ...
-2
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 ...
-1
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 $(...
