Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,364
questions
5
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1
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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 ...
4
votes
1
answer
108
views
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 ...
0
votes
1
answer
118
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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 ...
1
vote
0
answers
81
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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 ...
3
votes
2
answers
337
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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\...
9
votes
1
answer
349
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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 ...
11
votes
1
answer
579
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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 ...
3
votes
0
answers
76
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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 $(V,\rho,\Delta)$ ...
2
votes
1
answer
126
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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 ...
5
votes
1
answer
136
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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 ...
7
votes
1
answer
413
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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 ...
11
votes
1
answer
528
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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 ...
1
vote
0
answers
127
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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 ...
6
votes
1
answer
172
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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 ...
6
votes
1
answer
183
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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 ...
2
votes
0
answers
69
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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 $\...
1
vote
1
answer
122
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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 ${}^...
2
votes
2
answers
219
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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 ...
5
votes
1
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255
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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 ...
4
votes
1
answer
424
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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 ...
0
votes
1
answer
106
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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 &...
5
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0
answers
176
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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 ...
4
votes
1
answer
198
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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 ...
2
votes
0
answers
74
views
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 ...
5
votes
0
answers
108
views
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 ...
5
votes
1
answer
110
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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\...
3
votes
1
answer
141
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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/...
5
votes
0
answers
133
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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 ...
7
votes
1
answer
1k
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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 ...
7
votes
1
answer
200
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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 ...
11
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1
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418
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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.
...
5
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1
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333
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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 ...
6
votes
1
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138
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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 ...
7
votes
0
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172
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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 ...
6
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0
answers
117
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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 ...
2
votes
0
answers
69
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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 ...
10
votes
2
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876
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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 / \...
1
vote
0
answers
66
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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 ...
2
votes
0
answers
115
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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 ...
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, ...
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 ...
4
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0
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135
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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 $[-,-]$ ...
7
votes
1
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216
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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 ...
7
votes
1
answer
519
views
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 ...
7
votes
2
answers
371
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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 ...
5
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0
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198
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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 ...
4
votes
1
answer
179
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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 ...
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 \...
1
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0
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101
views
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. ...
1
vote
1
answer
88
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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 $a\geq 0$. ...