Questions tagged [ct.category-theory]

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

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9 votes
2 answers
287 views

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 ...
18 votes
1 answer
384 views

What is the group completion of finite sets with respect to cartesian product?

Let $\Sigma_+$ be the groupoid of finite pointed sets. The Barratt-Priddy-Quillen theorem tells us that the group completion of $\Sigma_+$ with respect to the symmetric monoidal structure given by ...
2 votes
1 answer
354 views

Closed embedding into a normal Hausdorff space and left lifting property

I am trying to understand the characterization of the class of closed embeddings into a normal Hausdorff space as the class of continuous maps satisfying the left lifting property with respect to a ...
4 votes
3 answers
780 views

Strictifying strong monoidal functors

Let $C_1$ and $C_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C_1 \rightarrow C_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal ...
7 votes
2 answers
1k views

What is a good definition of a mathematical structure?

At the moment I am writing a textbook in Foundations of Mathematics for students and trying to give a precise definition of a mathematical structure, which is the principal notion of structuralist ...
3 votes
0 answers
82 views

Preservation of Kan extensions

I am currently studying the theory of kan extensions more seriously, but I'm surprised of the apparent absence if theorems of preservation/reflection. What I have in mind is something along the lines ...
7 votes
0 answers
114 views

On Lemma 5.5.16 of Cisinski's "Higher Categories and Homotopical Algebra"

I have a question regarding Section 5 of Cisinski's "Higher Categories and Homotopical Algebra". Let us write $\mathbf{sSet}$ and $\mathbf{bisSet}$ for the categories of simplicial sets and ...
5 votes
0 answers
89 views

Points of the sheaf topos over Blass' category

There is a site $\textsf{Blass}$ used for (constructive) non-standard analysis, whose objects are sets equipped with a filter, and morphisms are continuous functions defined up to a small set. (It is ...
5 votes
1 answer
142 views

Proof that the inclusion $\Delta \to \mathbf{Pos}$ preserves colimits

I am trying to to prove that the inclusion $\Delta \to \mathbf{Pos}$ preserves colimits, where $\mathbf{Pos}$ is the category of partially ordered sets and monotone map, and $\Delta$ is the full ...
5 votes
1 answer
391 views

Does coproduct preserve cohomology in differential graded algebra category

Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...
8 votes
1 answer
472 views

Is there a better name for the "Mayer-Vietoris Octahedral axiom" and has it been studied?

$\newcommand{\K}{\mathcal{K}}$Say $\K$ is a triangulated category with suspension $\Sigma:\K\simeq \K$. In Iversen's book "Cohomology of Sheaves", he doesn't exactly examine triangulated ...
7 votes
1 answer
87 views

Bisimplicial spaces as a coequalizer of maps between "simpler" bisimplicial spaces

From a bisimplicial space $T$, one can consider the simplicial spaces $p \mapsto T_{pp}$, $p \mapsto | q \mapsto T_{pq}|$, and $q \mapsto |p \mapsto T_{pq}|$, where $| \cdot|$ denotes geometric ...
4 votes
4 answers
2k views

Homotopical Combinatorics

I have a question about the situation of homotopical combinatorics. There are many topics about combinatorial homotopy. But, I can't find any topic about homotopical combinatorics. More precisely, are ...
2 votes
0 answers
40 views

Godement product of lax 2-natural transformations

Which of the two obvious choices for the Godement product of lax $2$-natural transformations is ‘correct’? Specifically, recall that for natural categories, functors and natural transformations as ...
4 votes
1 answer
86 views

What are the 2-categorical mono/epimorphisms in the 2-category of relations?

$\newcommand{\procirc}{\mathbin{\diamond}}\newcommand{\rightproarrow}{\mathrel{\rightarrow\mkern-17mu|\mkern7mu}}$The monomorphisms in the 1-category $\mathsf{Rel}$ of sets and relations are precisely ...
0 votes
1 answer
159 views

Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically? [closed]

I asked this question on MSE here This question was inspired by: The influence of conjugacy class sizes on the structure of finite groups. My question is as follows: Is there a way to study the ...
9 votes
1 answer
257 views

$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$

This is a crosspost (with minor alterations). For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category ...
5 votes
1 answer
241 views

Quotients in categories of metric spaces

There are several categories whose objects are metric (or pseudo-metric) spaces. Natural choices of morphisms are continuous, uniformly continuous, Lipschitz or short (= non-expansive or contractive) ...
6 votes
0 answers
126 views

Does the localization functor $\mathcal{C}\to S^{-1}\mathcal{C}$ preserve finite colimits when $\mathcal{C}$ is not small? (size issues in proof)

$\def\colim{\operatorname{colim}} \def\hom{\operatorname{Hom}}$Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms of $\mathcal{C}$ that is a left multiplicative system. The ...
6 votes
1 answer
164 views

Variation on definition of logical functors avoiding power objects

Without power sets in meta-theory not every Grothendieck topos is an elementary topos, Set is still Grothendieck, but it lacks power objects. Now I am looking for a definition of a logical functor ...
16 votes
3 answers
2k views

Recommendations to learn about the use of toposes in logic?

I'd like to learn about the use of toposes in logic. The "logic" side I know quite well, but of the "topos" side I am totally ignorant. Which books/articles (formal and/or casual) ...
3 votes
0 answers
198 views

Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?

In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension". From [DLN, Theorem II (iii)], where the ...
27 votes
5 answers
9k views

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 ...
0 votes
1 answer
118 views

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 ...
9 votes
5 answers
944 views

English Reference for the Bénabou-Roubaud theorem

The Bénabou-Roubaud theorem links fibrational descent theory with monadicity. Particularly, it says that given a bifibration satisfying the Beck-Chevalley condition w.r.t some arrow $p$ in the base ...
3 votes
2 answers
240 views

Directed colimit of fully faithful functors

Suppose that for every $n\in\mathbb{N}$ we have a category $\mathcal{C}_n$ and a fully faithful functor $F_n:\mathcal{C}_n\hookrightarrow \mathcal{C}_{n+1}$. My question is whether fully faithful ...
0 votes
0 answers
68 views

Density of universe lifting functor in type theory

For some context, this is part of a larger story on relative monads, I have asked about a generalization of the Topos of coalgebras construction to the relative case here. The linked proposition 2.5 ...
8 votes
1 answer
154 views

Morphisms of hammocks in the simplicial localization

Let $\mathcal{C}$ be a category together with a wide subcategory $\mathcal{W} \subset \mathcal{C}$. In Calculating Simplicial Localizations by Dwyer and Kan, a morphism of hammocks is defined to ...
-1 votes
1 answer
137 views

Categories that admit all finite products but not all finite coproducts

What are examples for categories that admit all finite products but not all finite coproducts? (See also this question: Categories that admit all products but not all coproducts .)
4 votes
1 answer
211 views

Minus sign in rotated triangles in triangulated categories

Let $T$ be a triangulated category and $$ X \xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} X[1]$$ an exact triangle (or distinguished triangle). TR 2 implies that then the two rotated triangles $$...
2 votes
1 answer
157 views

Factorization systems for vector bundles

Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
8 votes
1 answer
186 views

Sequential colimit of iterated quotients of Cauchy sequences

We work in constructive mathematics. The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
4 votes
2 answers
310 views

Reference on Operads

I was reading 'Category for Scientists' by David Spivak and I'd like some references on operads with that kind of approach, using only "basic category theory", nothing too advanced. I ...
2 votes
0 answers
75 views

Implementation of the nerve of a category in GAP

I am trying to compute the second cohomology group of the coset poset $$\mathcal{C}_G\mathcal{F}=\{gA\mid g\in G,A\in\mathcal{F}\}$$ for a finite group $G$ and a family $\mathcal{F}$ of subgroups of $...
3 votes
0 answers
127 views

Is taking Freyd envelopes adjoint to taking stable module categories?

Let $T$ be an (idempotent-complete) triangulated category. Then the Freyd envelope $mod(T)$ is an abelian category, the universal recipient of a homological functor $T \to mod(T)$. The Freyd envelope ...
8 votes
0 answers
149 views

When is the Eilenberg-Moore category of a relative monad between two topoi a topos?

In the non-relative case, we have a theorem, that an Eilenberg-Moore category of algebras of a Monad $T$ on a topos is itself a topos if the monad in question has a right adjoint. Now how does this ...
11 votes
1 answer
617 views

Is the Thomason model structure on Cat simplicial? Is it a monoidal model category?

The Thomason model structure on the category of small categories is transferred from the Quillen model structure on simplicial sets along the right adjoint $Ex^2 \circ N$ (where $N$ is the nerve), i.e....
6 votes
2 answers
297 views

Is there a notion of a complex/analytic diffeological space?

I have a bit of a general question. This seems like something you can do, but I can't seem to find much reference for this.. Perhaps something like this already exists in a different guise. But, is ...
3 votes
7 answers
1k views

Categories that admit all products but not all coproducts

What are examples for categories that admit all products but not all coproducts.
3 votes
0 answers
90 views

Isomorphic objects have the same dimension (pivotal categories)

I want to prove that if two objects $X,Y$ in a pivotal category $\mathcal{C}$ are isomorphic, then $X$ and $Y$ have the same dimension, i.e., $$ \mathrm{dim}(X) = \mathrm{Tr}^{L}(\mathrm{id}_{X}) = \...
37 votes
7 answers
11k views

Limits in category theory and analysis

Is it possible to regard limits in analysis (say, of real sequences or more generally nets in topological spaces) as limits in category theory? Is there some formal connection? Edit ('13): Perhaps it ...
5 votes
1 answer
151 views

1-categorical universal properties for the smash product of pointed sets

Question I. Is the following statement, inspired by this one, true? Universal Property I. The symmetric monoidal structure on the category $\mathsf{Sets}_*$ of pointed sets and pointed maps between ...
3 votes
0 answers
35 views

Are the Drinfeld doubles of twist equivalent Hopf algebras twist equivalent?

Let $H_1$ and $H_2$ be finite dimensional Hopf algebras that are twist equivalent, i.e. $H_2$ is obtained from $H_1$ using a Drinfeld twist. My question is: are the Drinfeld doubles $D(H_1)$ and $D(...
1 vote
0 answers
42 views

Predicting coherence diagrams one dimension up

Assume we have a good working knowledge of $n$-dimensional category theory for some fixed $n$. It seems like it should be possible to 'predict' what coherence diagrams we're going to encounter in the ...
17 votes
5 answers
822 views

How can one characterise compactness-by-experiment?

There are a myriad different variations on the theme of "compactness", and some of them have even made it on to Wikipedia. I'm interested in finding out more about types of compactness that ...
3 votes
0 answers
50 views

Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions: Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...
5 votes
0 answers
75 views

Reciprocity for algebra objects in two algebraic categories

I think this question Compact Hausdorff and C^*-algebra "objects" in a category. shows that there is no reciprocity between categories of algebra-objects of two algebraic categories. So, ...
4 votes
1 answer
178 views

Presentability rank of tensor product of presentable categories

In this post category means $(\infty, 1)$-category. Let $X, Y$ be two presentable categories. We can then form their tensor product $X \otimes Y \cong \operatorname{ContFun}(X^{\mathrm{op}}, Y)$. Can ...
5 votes
1 answer
123 views

Co/limits and 2-co/limits of categories in the $\infty$- and $(\infty,2)$-category of $\infty$-categories

Recently, in a conversation with Gabriel, the following question came up: Question. Do co/limits of categories taken in the $\infty$-category of $\infty$-categories agree with the usual co/limits ...
12 votes
1 answer
178 views

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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