Questions tagged [ct.category-theory]

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

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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 ...
whatisandwhatshouldneverbe's user avatar
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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 ...
Keisuke Hoshino's user avatar
5 votes
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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 ...
Trebor's user avatar
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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 ...
Calin Tataru's user avatar
8 votes
1 answer
473 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 ...
FShrike's user avatar
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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 ...
Alec Rhea's user avatar
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4 votes
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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 ...
Emily's user avatar
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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 ...
Naif's user avatar
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$\operatorname{Spaces}/BG$ $\sim$ $\operatorname{Spaces}^G$ $\sim$ $??(\Omega G)$

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 ...
მამუკა ჯიბლაძე's user avatar
7 votes
1 answer
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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 ...
ems's user avatar
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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 ...
Elías Guisado Villalgordo's user avatar
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) ...
Jochen Wengenroth's user avatar
3 votes
2 answers
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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 ...
MikeTrooper's user avatar
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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 ...
Ilk's user avatar
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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 "...
wer's user avatar
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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 $...
Antoine's user avatar
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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 ...
Tim Campion's user avatar
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1 answer
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Factorization systems for vector bundles

Are there any well-known factorization systems for the category of vector bundles defined over topological spaces?
Nash's user avatar
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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 ...
Sebastien Palcoux's user avatar
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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}) = \...
NoetherNerd's user avatar
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 ...
Ilk's user avatar
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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 .)
Yilmaz Caddesi's user avatar
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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(...
Lagrenge's user avatar
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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 ...
Alec Rhea's user avatar
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3 votes
7 answers
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Categories that admit all products but not all coproducts

What are examples for categories that admit all products but not all coproducts.
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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” $\mathcal{D}_1$,...
hasManyStupidQuestions's user avatar
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 ...
Brendan Murphy's user avatar
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 ...
Emily's user avatar
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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 ...
Emily's user avatar
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2 votes
0 answers
85 views

Weighted limits and co-Yoneda

Is there a good reference that discusses weighted limits through the lens of the co-Yoneda embedding? Recall that the limit of a functor $F:\mathcal{C}\to{\bf Set}$ is canonically given by the set $${...
Alec Rhea's user avatar
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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 ...
Ilk's user avatar
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7 votes
0 answers
181 views

(2,1)-limits vs 2-limits of categories

In the section 1 of these notes, Emily de Oliveira Santos gives an explicit construction of most usual (co)limits in the 1-category of categories. In the next section she affirms that the same ...
Gabriel's user avatar
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10 votes
1 answer
367 views

Why can we take the colimit over the category of elements?

I'm trying to understand J. P. Murre's Tata notes on Grothendieck's theory of the fundamental group. For a Galois category $\mathcal C$ (which I'm taking to be locally small) with fundamental functor $...
themathandlanguagetutor's user avatar
11 votes
1 answer
371 views

Plus construction on Simplicial Sets?

I had asked this question in Math StackExchange a few days ago, but didn't get any answers. I believe its more suitable to be asked here. Write $\mathsf{sSet}$ for the category of simplicial sets and $...
wind's user avatar
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-1 votes
0 answers
38 views

What's a (single-sorted) algebraic signature? [migrated]

I am participating in an undergrad project that uses the book Nominal Sets by Andrew M. Pitts. I don't fully understand half of the things he attempts to explain but that is another issue, the main ...
Ata Berk Saraç's user avatar
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, ...
Nik Bren's user avatar
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0 answers
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Example of a pseudomonad on Cat whose pseudoalgebras are not the pseudoalgebras for a 2-monad

For every pseudomonad $T$ on the 2-category of (locally small) categories $\mathbf{Cat}$, we can consider the 2-category of $T$-pseudoalgebras and pseudomorphisms $T\text{-PsAlg}_p$, which is equipped ...
varkor's user avatar
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2 votes
1 answer
171 views

Is the category of simplicial $R$-modules closed monoidal?

I am trying to understand if the simplicial mapping space for simplicial $R$-Modules (or at least simplicial vector spaces) is adjoint to the (level-wise) tensor product. For simplicity, let me state ...
SetR's user avatar
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0 votes
0 answers
55 views

Names for product-like algebras involving a "duo of directed pseudoforests"

I am looking for the names (and/or for any information regarding) two algebras, one "free" and one "restricted" by an equivalence class. In both cases, there is an (infix) binary ...
user1661473's user avatar
4 votes
1 answer
83 views

Interpreting a diagram in Borceux-Quinteiro's paper on enriched sheaves

I am somewhat new to working with enriched categories, and have a question about how to interpret Definition 1.2 in Borceux-Quinteiro's paper A theory of enriched sheaves. The authors consider a ...
memento morison's user avatar
2 votes
2 answers
124 views

Infinite radical ideal cubed equals zero for tame hereditary Artin algebras

Let $A$ be a tame hereditary Artin algbera and mod$A$ the category of finitely generated (left) $A$-modules. Further, let rad$_A$ be the radical ideal of mod$A$, which is the smallest ideal containing ...
kevkev1695's user avatar
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8 votes
0 answers
483 views

In Mann's six-functor formalism, do diagrams with the forget-supports map commute?

One of the main goals in formalizing six-functor formalisms is to obtain some sort of "coherence theorem", affirming that "every diagram that should commute, commutes". In these ...
Gabriel's user avatar
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5 votes
0 answers
164 views

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 ...
Ilk's user avatar
  • 699
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 ...
varkor's user avatar
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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 ...
Morgan Rogers's user avatar
16 votes
2 answers
2k views

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 ...
David White's user avatar
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1 vote
0 answers
45 views

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: $\bullet \to \bullet$, ...
user234212323's user avatar
4 votes
0 answers
100 views

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 ...
user1077's user avatar
11 votes
2 answers
1k views

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 ...
h3fr43nd's user avatar
  • 221
1 vote
1 answer
113 views

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 ...
Elías Guisado Villalgordo's user avatar

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