# Questions tagged [ct.category-theory]

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

6,320
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### When are topoi of coalgebras atomic?

A geometric morphism is atomic if its inverse image is logical. Now consider a Grothendieck topos $\varepsilon$ and its terminal geometric morphism $\Gamma : \varepsilon \rightarrow Set$, topos is ...

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### Large V-categories admitting the construction of V-presheaves

By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I ...

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### Are there any interesting classes of limits containing finite limits?

Let $\Phi$ be a class of limit diagrams that contains all finite diagrams. Some examples include the classes $\Phi_{\kappa}$ of all diagrams of size bounded by a cardinal $\kappa$... but are there any ...

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### Is Freyd's thesis available online anywhere?

Peter Freyd is a great category theorist. His PhD dissertation, Functor Theory, dates from Princeton in 1960. It's cited as [14] in Mitchell's book Theory of categories. In fact, Google scholar says ...

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### Defining properties of categories out of an indicial category

$\newcommand{\Hom}{\operatorname{Hom}}$Suppose we want to define the category of arrows of $S$. Below are two forms of doing it.
Definition 1: If $D$ is of the following type: $\bullet \to \bullet$, ...

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### Localizations that are endofunctors

Usually when $F: C \rightarrow D$ is a localization functor, the categories $C$ and $D$ are not equivalent. My question is when is it possible for $C, D$ to be abstractly equivalent but $F$ is not an ...

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### Soft question: Deep learning and higher categories

Recently, I have stumbled upon certain articles and lecture videos that use category theory to explain certain aspects of machine learning or deep learning (e.g. Cats for AI and the paper An enriched ...

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### Reference request regarding faithfully exact functors between abelian categories

I am looking for a reference for the following result (or any subresult) in any book or notes:
Lemma. Let $F:\mathcal{A}\to\mathcal{B}$ be an exact functor between abelian categories. The following ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 $(V,\rho,\Delta)$ ...

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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 ...

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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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402
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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 ...

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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 ...

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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 ...

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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 $\...

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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 ${}^...

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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 ...

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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 ...

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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 ...

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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 &...

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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 ...

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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 ...

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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 ...

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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 ...

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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\...

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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/...

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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 ...

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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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### 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 ...

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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.
...

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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 ...

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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 ...

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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 ...

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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 ...

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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 ...

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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 / \...

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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 ...

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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 ...

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### 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, ...

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### 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 ...

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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 $[-,-]$ ...