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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.

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130 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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0answers
92 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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267 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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0answers
100 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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1answer
123 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 ...
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363 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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1answer
98 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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3answers
213 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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3answers
311 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 ...
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0answers
103 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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2answers
209 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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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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128 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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2answers
338 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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3answers
344 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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1answer
364 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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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 ...
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2answers
282 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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1answer
131 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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0answers
90 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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1answer
195 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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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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1answer
244 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 ...
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1answer
252 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 ...
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0answers
87 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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1answer
178 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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1answer
160 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 $(...
11
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1answer
203 views

Definitions of enriched monoidal category

This question is about two definitions of enriched monoidal categories I have: Let $\mathcal{V}$ be a symmetric monoidal closed category. The first definition: a $\mathcal{V}$-enriched category $\...
3
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1answer
174 views

Bijection between hom sets of equivalent categories?

I've been trying to prove the following claim, but am now unsure about its truth. Is it true, and if so, where can I find a proof? Claim: For any categories C, D, E such that C and D are equivalent, ...
4
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1answer
165 views

Does the notion of a compactly generated space (or $k$-space) depend on the choice of universe?

We recall the notion of a $k$-space (or compactly generated space) to fix our notations. For every topological space $X$, we can define a category $\mathfrak{M}_X$. The class of objects of $\mathfrak{...
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2answers
168 views

Contravariant internal hom

Let $\mathcal{C}$ a symmetric monoidal category. By using symmetry, it is very easy to show that the contravariant internal hom functor $[-,A]\colon\mathcal{C}^{op}\longrightarrow\mathcal{C}$ is ...
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1answer
166 views

The category of Multisets and Spans: morphism composition and tensor product

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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0answers
114 views

For which topological spaces does pullback along $\operatorname{ev}_0:B^I\to B$ have a right adjoint?

Let $B$ be a topological space. Consider the evaluation at zero of paths in $B$. This is a continuous map $\operatorname{ev}_0:B^I\to B$ where the domain carries the compact-open topology. For which ...
3
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1answer
131 views

Left lifting property and pushout

Let's say we are working in category $\mathcal{C}$, and that the three morphisms $ f: X \rightarrow X'$, $ g: Y \rightarrow Y'$ and $ h: Z \rightarrow Z'$ have the left lifting property with respect ...
3
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1answer
120 views

If the sum of a right (principal) ideal with a left one contains an invertible element and the product is zero then do they contain idempotents?

I am trying to solve a problem on additive categories, that gives the following question on (non-commutative unital associative) rings: if for elements $a$ and $b$ of a ring $R$ we have $ab=0$ and $a+...
11
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3answers
282 views

Quantum groups and deformations of the monoidal category of $U(\frak{g})$-modules

In the first answer for this question is writen, about the braided category of representation of the enveloping algebra $U(\frak{g})$, for $\frak{g}$ a semisimple Lie algebra: The space of ...
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0answers
143 views

How to visualize the Microsupport of a Sheaf?

I am looking through Persistent homology and microlocal sheaf theory to learn a bit on barcodes. They are require the notion of a microsupport of a sheaf, looks like it could be a rather concrete ...
4
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1answer
322 views

What is the definition of a $\mathcal{U}$-category?

Let $\mathcal{U}$ be a universe. The adaptation of the concept of a locally small category to universes is a $\mathcal{U}$-category. There are two definitions of $\mathcal{U}$ category I've met. $...
4
votes
1answer
363 views

Endomorphisms in the derived category

(Apologies if this question is trivial, but I'm way outside my area here.) Let $R$ be a commutative ring, $C^{\bullet}(R)$ the category of complexes of $R$-modules, and $D^{\bullet}(R)$ its derived ...
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0answers
108 views

DG functors along which contractions can be lifted

For an object $X$ in a DG category, its contraction is $r \in Hom^{-1}(X,X)$ such that $d(r)=1_X$. Let us say that contractions lift along a DG functor $F: \mathcal{C} \to \mathcal{D}$, if for a ...
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1answer
262 views

Understanding the functoriality of group homology

EDIT: I've decided to rephrase my question in order for it to be more concise and to the point. Let $G$ be a group, and let $F_\bullet\rightarrow\mathbb{Z}$ be a free $\mathbb{Z}[G]$-resolution of ...
5
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0answers
126 views

Quantum groups at $q=-1$

For a Drinfeld--Jimbo quantized enveloping algebra $U_q(\frak{g})$, it is standard knowledge that the categories of modules are very different in the $q$ a root of unity, and $q$ not a root of unity ...
4
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1answer
121 views

In a fibration, how do properties of arrows downstairs affect the base-change functors?

Fix a fibration of categories. Suppose $f:A\to B$ is an arrow in the base. What are the relations between the following pairs? $$f\text{ epi}\qquad f^\ast \text{ faithful}$$ $$f\text{ mono}\qquad f^...
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0answers
166 views

Testing for equivalences of $\infty$-categories on strictifications?

It is in general not too hard to show that maps between finite $CW$-complexes/finite simplicial sets are homotopy equivalences. Question : Can we do something similar for: quasi-categorical ...
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3answers
219 views

Constructive proof of exponential law in a category

I'm trying to write a constructive proof of the isomorphism ${Z}^{X \times Y} \equiv (Z^Y)^X$ in a category with exponential objects. I can construct a map $(Z^Y)^X \rightarrow {Z}^{X \times Y}$, but ...
3
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0answers
161 views

Tannaka duality for $DG$ indschemes

In Lurie's paper on Tannaka duality for geometric stacks he proves that there is a natural isomorphism $$\operatorname{Hom}(X,Y) \cong \operatorname{Hom}(\mathbf{QC}(Y), \mathbf{QC}(X)$$ where $X$ and ...
3
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1answer
67 views

sequences of iterated orthogonals (lifting property) in a category

I am looking for examples of properties of morphisms defined by taking orthogonals with respect to the Quillen lifting property. For example, several iterated orthogonals of $ \emptyset\...
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1answer
90 views

Equivalent conditions to be a lax idempotent contravariant monads

$\require{AMScd}$As already said, a contravariant monad is "like a monad, but a contravariant functor": the multiplication and unit are dinatural transformations an algebra is a map $a : TA\to A$ ...
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1answer
129 views

Cellularity of anodyne extensions?

Are the anodyne extensions of simplicial sets always relative cell complexes of horn inclusions? (i.e. there is no need to consider retracts) If not, is there a known counterexample? Similarly, does ...
21
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1answer
269 views

Explicit Left Adjoint to Forgetful Functor from Cartesian to Symmetric Monoidal Categories

There is a forgetful functor from the category of (small) Cartesian monoidal categories (a symmetric monoidal category in which the tensor product is given by the categorical product) to the category ...