Questions tagged [ct.category-theory]

Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.

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When would a left admissible triangulated subcategory be admissible

I'm walking through the proof of [1, Thm 16 at pp. 515] and am stuck at the first sentence after equation (12), where the author states that the decomposition (12) is semiorthogonal when $a\geq 0$. ...
Noto_Ootori's user avatar
3 votes
0 answers
114 views

Tannaka duality for Hopf algebroids

Setting. Let $k$ be a field, $A$ a finite-dimensional $k$-algebra, and $H$ a Hopf algebroid over $A$ with invertible antipode. Denote by $\operatorname{mod}(H)$ the category of finite-dimensional ...
Max Demirdilek's user avatar
27 votes
5 answers
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Why does mathematics seem to have a polarity bias?

Why does mathematics seem to have a polarity bias, i.e., why are products more common than coproducts, algebras more common than coalgebras, limits more common than colimits, monads more common than ...
Cameron Zwarich's user avatar
4 votes
1 answer
174 views

Monoidal topology and coarse spaces

Is there a description of (quasi-)coarse spaces that is analogous to the description of (quasi-)uniform spaces as lax algebras?
Cameron Zwarich's user avatar
1 vote
0 answers
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Site structure on smooth fibered manifolds

Let $\mathsf{FB}_{s}$ be the category whose objects are smooth fibered manifolds, and whose morphisms are smooth strong projectable maps. Recall that given fibered manifolds $(\pi:Y\rightarrow X)$ and ...
Bence Racskó's user avatar
2 votes
1 answer
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Is uniform dimension monotonic in quotients when there is a unique indecomposable injective?

The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $\mathcal C$. Recall ...
Tim Campion's user avatar
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Are fibrations of small categories fibrations?

The isofibrations are the fibrations of the canonical model structure of the category of small categories. If I call fibration of small categories the same notion by removing the word isomorphism, i.e....
Philippe Gaucher's user avatar
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Existence and characterisations of left Kan extensions and liftings in the bicategory of relations II

This is the second part to a previous question regarding left Kan extensions/lifts in the bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations, which has now been split into two ...
Emily's user avatar
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Categorical description of umbral calculus?

The theme of my current research project is related to umbral calculus in the context of more general algebraic structures, like Hopf algebras (and, in particular, shuffle algebras), so I am trying to ...
Daigaku no Baku's user avatar
13 votes
1 answer
612 views

The category theoretic origin of arithmetic product

$\newcommand\Bij{\mathrm{Bij}}\newcommand\Set{\mathrm{Set}}\newcommand\Species{\mathrm{Species}}$The paper "On the arithmetic product of combinatorial species" by Maia and Méndez introduces ...
fosco's user avatar
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Does every $\kappa$-compact topos embedd relatively $\kappa$-tidily into a presheaf topos?

Let $\kappa$ be a regular cardinal and say that a topos $\mathcal{E}$ is $\kappa$-compact if the global sections $\gamma_{\ast} : \mathcal{E} \to \mathsf{Set}$ preserves $\kappa$-filtered colimits. My ...
David Jaz Myers's user avatar
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Why equaliser of product and terminal object is coproduct?

I’m reading “Sheaves in geometry and logic”, in page 80: Please refer to [1]: https://i.stack.imgur.com/INrU0.jpg It says “…,therefore $FU=\coprod_{x\in U} fx$. The space…”. So could anyone please ...
SuBonan's user avatar
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Drinfeld center of non-rigid closed monoidal categories

Context. The Drinfeld center of a rigid monoidal category $\mathcal{C}$ is again rigid. This is not hard to see: Given an object $X\in \mathcal{C}$ together with a half-braiding $\phi_X:X\otimes-\...
Max Demirdilek's user avatar
11 votes
1 answer
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What is the commutative coproduct and where can I learn more about it?

This is a reformulation of a Mathematics Stack Exchange question that got no answer there, and, at the same time, a follow-up to another question on MSE. The original problem was to prove $U(\mathfrak{...
Daigaku no Baku's user avatar
7 votes
0 answers
248 views

What exactly is a Tannakian subcategory?

I've searched all the standard references (Deligne--Milne, Saavedra-Rivano) and cannot find a definition of Tannakian subcategory. What I find is many authors who discuss the Tannakian subcategory ...
David Corwin's user avatar
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Has anyone studied factoring as a CO-product?

In factorization, like integer factorization, you start with an integer and end up with a kind-of list of pairs of other elements, namely the factors. I want to explore the "Co-ness" of this....
Ben Sprott's user avatar
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Is this an instance of the snake lemma?

I recently had need of the following fact (in the category of abelian groups, but I'm pretty sure it holds for all abelian categories): given a commutative diagram of the form (quiver link), thus $k \...
Terry Tao's user avatar
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Coevaluation for linear categories

For a field $k$ and an associative $k$-algebra $R$, the $k$-linear category $R\operatorname{-Mod}$ is self dual inside $\operatorname{DGCat}_k$, with the counit map sending $k$ to $R$ regarded as a ...
E. KOW's user avatar
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The algebras and coalgebras of the homology functor

My question is very simple, but I suspect far from the intuition with which singular homology is introduced. Consider singular homology as a functor $$H_n : {\sf Top}\times{\sf Ab} \to \sf Ab$$ This ...
fosco's user avatar
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2 votes
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Using the mapping cone to show that a chain map defines a stable equivalence between two symmetric algebras

This question is about an argument in the proof of Theorem 9.8.8 in Linckelmanns Block Theory of Finite Group Algebras. I need to understand the argument in order to do something similar in my ...
jb2g4's user avatar
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What conditions on an Abelian category allow members of a direct sum to be determined entirely by their components?

EDIT: In comments, with thanks to Maxime Ramzi, this question has a good answer in that what I want to be true is true when $\mathscr{A}$ satisfies axiom $\mathsf{AB}5$, that $\mathscr{A}$ is ...
FShrike's user avatar
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3 votes
1 answer
163 views

Nontrivial example of when monadic functors don't compose

It is well-known that the composite of monadic functors $U: C \to C'$ and $U': C' \to C''$ need not be monadic. One standard example is the forgetful functor $\mathrm{Cat} \to \mathrm{RefGph}$ from ...
Todd Trimble's user avatar
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Existence and characterisations of left Kan extensions and liftings in the bicategory of relations I

The bicategory $\mathsf{Rel}$ of sets, relations, and inclusions of relations has right Kan extensions and right Kan lifts¹, however I believe it does not have all left Kan extensions/lifts. Is it ...
Emily's user avatar
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4 votes
1 answer
233 views

On the initiality of the inclusion from the simplex category to the paracycle category

Thm B.3 of Nikolaus and Scholze shows that the natural inclusion $\Delta \to \Lambda_\infty$, from the simplex category to the paracycle category, is an initial functor, i.e. satisfies the hypotheses ...
Tim Campion's user avatar
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2 votes
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234 views

Existence and explicit descriptions for left and right Kan extensions and lifts in bicategories of spans

Given a category $\mathcal{C}$, it follows from Proposition 4.1 of Day's Brian Day, Limit spaces and closed span categories, Lecture Notes in Mathematics, 420, 1974 (doi:10.1007/BFb0063100). That ...
Emily's user avatar
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6 votes
1 answer
526 views

In an abelian category with no nontrivial Serre subcategories, does every short exact sequence split?

Question: Let $\mathcal A$ be an abelian category. Suppose that the only Serre subcategories of $\mathcal A$ are the zero category and $\mathcal A$ itself. Does it follow that every short exact ...
Tim Campion's user avatar
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6 votes
1 answer
306 views

$\infty$-topos as an internal $\infty$-category in itself

I'm interested (both autonomously and directly related to my work) in the natural internalization of $\infty$-topos sheaves in it (as usual, assuming Grothendieck universes). Is there any literature ...
Arshak Aivazian's user avatar
5 votes
2 answers
394 views

How is the classification of groups extensions by $H^2$ related to Yoneda Ext?

It is well-known that group extensions $$1\to A \to H \to G \to 1$$ where $A$ is abelian with a $G$-action such that the conjugation action of $G$ on $A$ agree with this fixed action are classified ...
Antoine Labelle's user avatar
2 votes
1 answer
279 views

Hypercover and hyper descent

I am trying to understand the descent condition using hypercovers. The condition says that a hyper cover of a scheme $X$ is a simplicial set $Y_{\bullet}$ that satisfies the condition $Y_n\rightarrow ...
Hello's user avatar
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4 votes
1 answer
587 views

Coherent sheaves, Serre’s theorem and ext groups

Let $X$ be a smooth projective variety over an algebraically closed field $k$ (if necessary we assume that $\operatorname{ch}(k)=0$). Let $O_X(1)$ be a very ample invertible sheaf on $X$. Then, the ...
Walterfield's user avatar
12 votes
0 answers
238 views

Intuitionistic proofs of propositional formulae versus natural transformations between finite sets

The setup: Given a formula $\varphi$ of intuitionistic propositional logic (i.e., made from the connectors $\Rightarrow$, $\land$, $\lor$, $\top$ and $\bot$ from propositional variables $A,B,C,\ldots$)...
Gro-Tsen's user avatar
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2 votes
0 answers
114 views

Universal property of Isbell duality

Let's take $\mathrm{C}$ be a category, let's have an adjunction $(\mathcal{O} \dashv \operatorname{Spec}) : \mathrm{CoPresheaves(C)} \leftrightarrows \mathrm{Presheaves(C)}$. One such adjunction is ...
Ilk's user avatar
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3 votes
0 answers
84 views

Tensor product of functors, central Hopf monad and star-autonomy

Setting. Let $\mathcal{C}$ be a category and $(\mathcal{V},\otimes,I, \multimap)$ a (symmetric) closed monoidal category. Let $F:\mathcal{C}\rightarrow \mathcal{V}$ be a functor and $X\in \mathcal{V}$ ...
Max Demirdilek's user avatar
2 votes
0 answers
91 views

Are covering families of localizations stable under pushouts?

For a commutative ring $A$, we call a finite family of localizations $A \to A_{S_i}$ (where $S_i$ are some subsets of $A$) a covering if the canonical morphism $A \to \prod A_{S_i}$ is an effective ...
Arshak Aivazian's user avatar
7 votes
1 answer
167 views

Comonoid homomorphisms in the bicategory of profunctors

Cartesian bicategories axiomatize the intuitively evident but mathematically elusive "cartesian" product on bicategories such as Rel, Span, and Prof. An important concept for cartesian ...
Evan Patterson's user avatar
5 votes
1 answer
353 views

Does the simplex map to the cube?

For every $n \geq 0$ there is an inclusion of the ordered set $\{0<1<\dots<n\}$ into the product $\{0<1\}^{\times n}$ sending $i$ to the increasing sequence $(0 < \dots<0<1<\...
willie's user avatar
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2 votes
0 answers
139 views

Infinitesimal criteria for unramified morphism on stacks

In an Artin stack, the tangent complex is fundamentally a derived object, often viewed as a 2-term complex via some quotient presentation, as discussed in Raskin's note here, for example. In ...
C.D.'s user avatar
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2 votes
0 answers
141 views

Trying to decode a module functor

This is section 5.1 from, ARITHMETIC DERIVATIVES THROUGH GEOMETRY OF NUMBERS by Hector Pasten. Let $A$ be a commutative unitary ring, let $R$ be a commutative monoid, and let $\alpha : R \to A$ be a ...
Ilk's user avatar
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5 votes
1 answer
307 views

A result on symmetric closed monoidal categories

In this discussion from the categories mailing there is mention of the following result by Robin Houston, supposedly proved in 2006: Theorem. Let $\mathcal{C}$ be a symmetric closed monoidal category,...
Max Demirdilek's user avatar
8 votes
0 answers
124 views

Locally presentable and accessible categories without the axiom of choice?

Is there a good reference for the study of locally presentable and accessible categories without the axiom of choice? For instance, it seems one will need to understand: What is a good notion of $\...
Tim Campion's user avatar
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5 votes
1 answer
182 views

3-functoriality of the lax Gray tensor product

In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For ...
varkor's user avatar
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1 vote
0 answers
194 views

Kan extensions in Grothendieck school

Considering both the ubiquity of Kan extensions in category theory (as MacLane stated, 'The notion of Kan extensions subsumes all the other fundamental concepts of category theory.'), its early ...
user234212323's user avatar
4 votes
1 answer
238 views

Is there a category theoretic definition of a cryptographic commitment scheme?

I'm trying to come up with a composeable framework for cryptographic commitment schemes, where inclusion proofs can be combined in different ways. I'm thinking this can be done with category theory, ...
eryb's user avatar
  • 153
3 votes
1 answer
142 views

What are the internal adjunctions in the bicategory $\mathsf{Span}$?

Recently I've been trying to understand spans better, in particular how they relate to relations, as both may be thought of as "multivalued functions between sets" (see Bruni and Gadducci - ...
Emily's user avatar
  • 10.3k
2 votes
1 answer
114 views

Duality in a monoidal category as a functor

In a rigid monoidal category $\mathcal{M}$ every object has a (say left) dual. Is the process of taking duals functorial? More specifically - is there a well-defined functor $$ \mathcal{M} \to \...
Yilmaz Caddesi's user avatar
4 votes
0 answers
535 views

$\mathbb{Z}[T]$-Solidification in light condensed setting

In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...
Jonas Heintze's user avatar
3 votes
1 answer
257 views

A question about rigid objects in monoidal categories

Let $(\mathcal{A},\otimes,1_{\mathcal{A}})$ be a monoidal category, and let $M$ be a rigid object of $M$, with left and right dual respectively denoted by $$ \Big\{M^*,~~~ \epsilon_l:M^* \otimes M \to ...
Yilmaz Caddesi's user avatar
3 votes
1 answer
250 views

Do objects in the derived category behave stackily?

It is well known that derived categories (I'm particularly thinking of constructible derived categories and derived categories of D-modules) don't form a stack. In particular given morphisms in the ...
l-r-b's user avatar
  • 85
2 votes
1 answer
81 views

Are the injections of a coproduct a cover in the canonical pretopology?

Assume we're in a category $C$ with all pullbacks and finite coproducts. Recall that the canonical coverage of $C$ is the finest Grothendieck (pre) topology for which all representables are sheaves. A ...
Joey Eremondi's user avatar
2 votes
0 answers
128 views

Tensor product of objectwise weak homotopy equivalences of $\mathcal{M}$-spaces

I consider the enriched category $[\mathcal{M}^{op},\mathrm{Top}]$ of enriched functors (I call them $\mathcal{M}$-spaces) from the enriched small category $\mathcal{M}^{op}$ to the enriched category $...
Philippe Gaucher's user avatar