Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,365
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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 $a\geq 0$. ...
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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 ...
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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 ...
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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?
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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 ...
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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 ...
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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....
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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 ...
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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 ...
13
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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 ...
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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 ...
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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 ...
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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-\...
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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{...
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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 ...
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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....
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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 \...
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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 ...
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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 ...
2
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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 ...
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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 ...
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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 ...
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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 ...
4
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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 ...
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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 ...
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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 ...
6
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$\infty$-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 ...
5
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394
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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 ...
2
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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 ...
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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 ...
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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 $A,B,C,\ldots$)...
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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 ...
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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 $X\in \mathcal{V}$ ...
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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 ...
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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 ...
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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<\...
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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 ...
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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 ...
5
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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,...
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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 $\...
5
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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 ...
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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 ...
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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, ...
3
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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 - ...
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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
$$
\mathcal{M} \to \...
4
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$\mathbb{Z}[T]$-Solidification in light condensed setting
In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...
3
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A question about rigid objects in monoidal categories
Let $(\mathcal{A},\otimes,1_{\mathcal{A}})$ be a monoidal category, and let $M$ be a rigid object of $M$, with left and right dual respectively denoted by
$$
\Big\{M^*,~~~ \epsilon_l:M^* \otimes M \to ...
3
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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 ...
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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 ...
2
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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 $...