Questions tagged [ct.category-theory]

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

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Completeness of comma $\infty$-categories

Let $\mathsf{A},\mathsf{B},$ and $\mathsf{C}$ be (ordinary) categories and $F : \mathsf{A}\to\mathsf{C}$ and $G : \mathsf{B}\to\mathsf{C}$ be functors such that $\mathsf{A}$ and $\mathsf{B}$ are ...
Stahl's user avatar
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Does the Gray tensor product exhibit Gray as a monoidal Gray-category?

Crans constructs a lax Gray tensor product for strict omega-categories that defines a monoidal structure on the category of strict omega-categories $\omega Cat$. We write Gray for this monoidal ...
willie's user avatar
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The generating series of the weighted species of fixpoints

I am wondering if the series $$\sum_{n=0}^\infty \left(\sum_{k=0}^n \frac{D_{n-k}}{k!(n-k)!}t^k\right)X^n$$ where $D_m$ is the number of derangements of $m$ letters, admits a representation in closed ...
fosco's user avatar
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Fibre product and submersion of PL-manifolds

Let us consider the fibre product $M\times_{f,g} M'$ of $M \xrightarrow{f} N \xleftarrow{g} M'$. If $M,M'$ and $N$ are smooth manifolds and $f$ is a submersion, then $M\times_{f,g} M'$ is again a ...
KoopaTroopa's user avatar
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2 answers
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$R$-Module objects in cartesian closed categories

I am looking for a reference for the following statement. Theorem. Let $C$ be a regular, well-powered, countably complete cartesian closed category, $R$ be a (commutative) ring object in $C$, $R\...
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Characterize algebras of the "topological simplices" operad

The operad of topological simplices, which I'll denote $\Delta$, has as $n$-ary operations the set $$ \Delta_n:=\left\{P\colon\{1,\ldots,n\}\to[0,1]\;\middle|\;1=\sum_{i=1}^n P(i)\right\} $$ of ...
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On the classification of second-countable Stone spaces

Let $X$ be a Stone space (i.e. totally disconnected compact Hausdorff). Then the following are equivalent: $X$ is second countable $X$ is metrizable $X$ has countably many clopen subsets $X$ is an ...
Tim Campion's user avatar
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Yetter-Drinfeld modules for Hopf monads

1. Context. 1.1. Classical Yetter-Drinfeld modules. Let $H$ a bialgebra in a braided monoidal category $\mathcal{C}$. A left-right Yetter-Drinfeld module over $H$ is a triple $(V,\rho,\Delta)$ ...
Max Demirdilek's user avatar
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The separability of superextensions

The superextension $\lambda X$ of a compact Hausdorff space $X$ is the space of maximal linked systems of closed subsets of $X$, endowed with the Vietoris topology inherited from the double hyperspace ...
Taras Banakh's user avatar
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One-object lax natural transformation

A lax natural transformation $\alpha$ between two pseudofunctors $F,G: \mathcal{K} \to \mathcal{L}$ between bicategories $\mathcal{K}, \mathcal{L}$ consists of the following data: For every object $A ...
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Model categories as a tool to resolve size issues for localizing categories

I have a rather basic question about one motivation for introducing model categories in David White's notes, as a possible way to overcome troubles with size issues appearing when localizing a ...
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Can you deduce the correspondence between 2D oriented TQFTs and commutative Frobenius algebras from the (framed) Cobordism Hypothesis?

Background I am currently writing an MSc dissertation on TQFTs (and Khovanov homology, but that is unrelated to this question). After having read most of Kock's book on the equivalence between 2D ...
Santiago Pareja Pérez's user avatar
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Piecewise construction of a functor from an $(\infty,1)$-category with an orthogonal factorization system

For the simpler case of 1-categories, consider a 1-category $C$ and an orthogonal factorization system $(L,R)$ on $C$. Let $C_L$ and $C_R$ denote the wide subcategories of $C$ corresponding to the ...
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Hopf monads in categorical probability theory

1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really ...
Max Demirdilek's user avatar
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Finitely presentable objects in the categories of algebras of $\infty$-algebraic theories

By default, all terms are understood in the infinity sense (“category” means “$(\infty, 1)$-category”, etc.). An object $A$ in a category is said to be finitely presentable (or compact) if the functor ...
Arshak Aivazian's user avatar
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Colimits from van Kampen cocones

Let $\mathcal{C}$ be a category with pullbacks, $\mathcal{J}$ a small category, $F : \mathcal{J} \to \mathcal{C}$ a diagram and $\kappa : F \Rightarrow \Delta X$ a cocone in $\mathcal{C}$. Let $\...
Naïm Favier's user avatar
1 vote
1 answer
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Dual objects in an abelian monoidal category

Let $(\mathcal{M},\otimes)$ be a monoidal category that is also abelian, and assume that $\otimes$ is bilinear with respect to the biproduct. Let $X$ and $Y$ be two objects that admit (left) duals ${}^...
Yilmaz Caddesi's user avatar
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2 answers
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Is the mapping cylinder a replacement for morphism by cofibration in model categories?

Let $M$ be a model category, consider a very good cylinder object $X \coprod X \to X \times I \overset{\operatorname{pr}}{\to} X$ (here $X \times I$ is just a notation, no object $I$ is implied), that ...
Arshak Aivazian's user avatar
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Are Euclidean spaces $\Delta$-generated?

From the definition of $\Delta$-generated it seems like $\mathbb R$ should be $\Delta$-generated, as $\mathbb R$ is final with respect to all continuous maps $\mathbb R^n \to \mathbb R$. However, the ...
William B.'s user avatar
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1 answer
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Exact sequences in Positselski's coderived category induce distinguished triangles

I am learning about Positselski's co- and contraderived categories. We know that short exact sequences do not generally induce distinguished triangles in the homotopy category but they do in the usual ...
So Let's user avatar
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Is every locally $\kappa$-presentable category, also locally $\tau$-presentable for any $\tau > \kappa$?

Let $\kappa$ be a small regular cardinal and $D$ a locally $\kappa$-presentable category. Is it true that $D$ is also locally $\tau$-presentable for any $\tau > \kappa.$ Adamek und Rosicky show in &...
willie's user avatar
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Is there a way to “derive” a (non-exact) functor which preserves images?

Let $F : \mathcal A \to \mathcal B$ be an additive functor between abelian categories. If $F$ is left exact, then under certain conditions $F$ admits right derived functors which “measure” its failure ...
Tim Campion's user avatar
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Is every compact, sober, second-countable space the image of $2^\omega$?

As a bonus, is every compact, $T_0$, second-countable space the image of $2^\omega \times \omega$? As a further bonus, can we strengthen "image" to "quotient"? My motivation for ...
Robin Saunders's user avatar
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What can be said about the free-forgetful adjunction of monad algebras with respect to topoi?

For a monad T on a topos E, if T has a right adjoint, then the Eilenberg-Moore category of algebras of T is equivalent to the co-Eilenberg-Moore category of co-algebras for the right adjoint comonad ...
Ilk's user avatar
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Does the forgetful functor from Lie algebroids to vector bundles have a right adjoint

Let $\mathcal{S}$ be an appropriate category of smooth spaces, and let $\mathrm{Vect} \colon \mathcal{S}^\mathrm{op} \to \mathsf{Cat}$ be the functor sending a space to the category of vector bundles ...
cheshircat's user avatar
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Relations with "for each" composition and its properties (coming from profunctors with end composition)

$\newcommand{\sq}{\mathbin{\square}}$The usual composition $S\diamond R$ of two relations $R\colon A⇸B$ and $S\colon B⇸C$ is defined as follows: For each $(a,c)\in A\times C$, we declare $a\sim_{S\...
Emily's user avatar
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3 votes
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Kernels and cokernels in a quotient of an abelian category

I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
Ji Woong Park's user avatar
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Classifying the algebraic structure on endomorphism sets

This is motivated by defining modules in a general sense, which is an appropriate homomorphism from $R$ to $\textrm{End}(X)$. If $X$ comes from different categories, the endomorphism set will have ...
Trebor's user avatar
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Which revolutions in topology and geometry can we expect in the next 20 years? [closed]

In my limited perspective on the history of mathematics, I can name at least two big revolutions in Topology and Geometry (broadly construed): the introduction of Schemes in Algebraic Geometry, and ...
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1 answer
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Compact objects in slice categories of finitely presentable categories

Given a locally finitely presentable category $\mathscr C$ and an object $X \in \mathscr C$, it is not so hard to show that a morphism $(X \to Y)$ is compact in $\mathscr C_{X/}$ if it can be obtained ...
R. van Dobben de Bruyn's user avatar
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1 answer
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Are flat functors out of a finite category necessarily finite?

Note: I've originally asked this question on math stack exchange, but I have learnt that this is the better place to ask for research level questions, so I have deleted the original question there. ...
Lingyuan Ye's user avatar
5 votes
1 answer
333 views

Day convolution and sheafification

$\DeclareMathOperator\Psh{Psh}\DeclareMathOperator\Sh{Sh}\newcommand\copower{\mathrm{copower}}$I was looking through Bodil Biering's thesis On the Logic of Bunched Implications - and its relation to ...
Anthony D'Arienzo's user avatar
6 votes
1 answer
138 views

Does the 2-category of double categories and vertical transformations have flexible limits?

Consider the 2-category of pseudo-double categories (with the weak composition in the horizontal direction and the strict composition in the vertical direction), strong double functors, and vertical ...
David Jaz Myers's user avatar
7 votes
0 answers
172 views

Why are Gabriel categories closed under localization?

Let $\mathcal K$ be a Grothendieck category. Recall the Gabriel filtration $0 \subseteq \mathcal K_1 \subseteq \cdots \mathcal K$ of localizing subcategories, where $\mathcal K_{\alpha+1}$ is ...
Tim Campion's user avatar
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6 votes
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Looking for Lawvere's "Closed categories and biclosed bicategories" lecture notes

The nLab page on closed bicategories reads Closed bicategories were introduced by Lawvere in unpublished lecture notes Closed categories and biclosed bicategories (1971). This work has also been ...
Emily's user avatar
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2 votes
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Recover the $C_k$-action of a cyclic object as from the $S^1$-action on Hochschild chain

$\newcommand{\Fun}{\operatorname{Fun}}$Let $X\in\Fun(\Lambda^\mathrm{op}, \mathcal{C})$ be a cyclic object in the ($\infty$-)symmetric monoidal category $\mathcal{C}$, where $\Lambda$ is the cyclic ...
Bingyu Zhang's user avatar
10 votes
2 answers
876 views

Example of a Grothendieck category which is not Gabriel?

Following Gabriel, for $\mathcal C$ a Grothendieck category, set $\mathcal T(\mathcal C)$ to be the localizing subcategory generated by the objects of finite length, and $\mathcal C' = \mathcal C / \...
Tim Campion's user avatar
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1 vote
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Beck's original formulation of the precise tripleability theorem. Reference when considering reflexive pairs?

Thanks to MO's user Varkor, we have access to Beck's original untitled manuscript where Beck first stated his precise tripleability theorem. Up to terminological isomorphism, the PTT as stated in p. 3 ...
Elías Guisado Villalgordo's user avatar
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Left duals and right duals are also isomorphic in a semisimple category

In the n-Lab page https://ncatlab.org/nlab/show/rigid+monoidal+category it is written that Left duals and right duals are also isomorphic in a semisimple category. For a left dual semisimplicity ...
Yilmaz Caddesi's user avatar
2 votes
1 answer
536 views

Logical content of Gauss's Lemma (arithmetic)

In the context where $a$, $b$, $c$ are integers we have $(a \mid bc, a\land b = 1) \Rightarrow a\mid c$. This result is called Gauss's Lemma in French Highschool. It is well known that (Steve Awodey, ...
smed's user avatar
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13 votes
2 answers
945 views

Categories on which one can determine all model structures?

Famously, there are exactly nine model structures on the category of sets, which are detailed here. In this case, one can exhaustively determine all six weak factorization systems and then see which ...
Tim Campion's user avatar
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4 votes
0 answers
135 views

Examples of $\ast$-autonomous $(\infty,1)$-categories

A $\ast$-autonomous category is a biclosed monoidal category together with a dualizing object. An object $\bot$ in a biclosed monoidal category $(\mathcal{C},\otimes)$ with left internal hom $[-,-]$ ...
Max Demirdilek's user avatar
7 votes
1 answer
216 views

Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?

For ordinary category theory, we have the following fact. A weighted colimit of a functor can always be equivalently expressed as a colimit of a different functor. Specifically, the weighted colimit ...
Nick Hu's user avatar
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1 answer
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Is there any categorical version of central limit theorem?

I'm not sure if the question even makes sense, but I wonder if there's any categorical reason that explains importance of Gaussian/normal distribution. In the ordinary probability theory, I guess ...
Seewoo Lee's user avatar
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7 votes
2 answers
371 views

Natural ways to make a functor adjoint

Let $F: C \to D$ be a functor between two categories without a right adjoint. What are some natural ways to create a right adjoint for $F$? Of course, this does not make sense on the nose. One needs ...
Student's user avatar
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5 votes
0 answers
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Classical first-order model theory via hyperdoctrines

I have been reading this discussion by John Baez and Michael Weiss and I find this approach to model theory using boolean hyper-doctrines very interesting. One of their goal was to arrive at a proof ...
Antoine Labelle's user avatar
4 votes
1 answer
179 views

Are there (non Lagrangian) algebras of Turaev-Viro TQFTs which cannot be completed to Lagrangian algebras?

Consider a 3d TQFT of the Turaev-Viro type, say TV$(\mathcal{C})$, where $\mathcal{C}$ is some fusion category. Equivalently, this is a TQFT admitting Lagrangian algebra objects $\mathcal{L}$ of the ...
Andrea Antinucci's user avatar
8 votes
1 answer
210 views

Can finite presentability be tested with respect to sequential colimits?

Let $\mathcal C$ be a locally finitely-presentable category, and let $C \in \mathcal C$ be an object such that for all sequential colimits, the map $$\varinjlim \operatorname{Hom}(C, X_i) \to \...
Tim Campion's user avatar
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1 vote
0 answers
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When is a container a monad?

The category of polynomial functors on Set is equivalent to the category of containers. We have a prescription for when a container is a comonad. There are a few other questions that come to mind. ...
Ben Sprott's user avatar
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1 vote
1 answer
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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