Questions tagged [ct.category-theory]

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

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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 $${...
6 votes
1 answer
282 views

When does $\mathbf{Top}/X$ embedd fully faithfully into $\mathbf{Top}$?

Under what conditions on the topological space $X$ is the overcategory $\mathbf{Top}/X$ of topological spaces over $X$ equivalent to a full subcategory of $\mathbf{Top}$? Surely if $X$ terminal i.e. a ...
6 votes
3 answers
321 views

Poisson and homotopy Poisson operads

$\newcommand{\Pois}{\mathit{Pois}}\newcommand{\Comm}{\mathit{Comm}}\newcommand{\Lie}{\mathit{Lie}}$For my thesis, I'm trying to understand the Poisson operad (I'll call it $\Pois$) and its homotopy ...
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 ...
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 $...
26 votes
2 answers
7k views

Large cardinal axioms and Grothendieck universes

A cardinal $\lambda$ is weakly inaccessible, iff a. it is regular (i.e. a set of cardinality $\lambda$ can't be represented as a union of sets of cardinality $<\lambda$ indexed by a set of ...
13 votes
2 answers
720 views

Can one recover an algebraically closed field $k$ from the dots and arrows of its category of finitely generated $k$-algebras?

You are gracious enough to host me for a few days while I attend a conference. After I leave, you're surprised to see a gift on the kitchen table. It's a box with a category inside! The objects aren't ...
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 $...
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 ...
-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 ...
7 votes
1 answer
216 views

Locating the typed version of Hoàng Xuân Sính's thesis on Gr-categories

Hoàng Xuân Sính was a Vietnamese student of Grothendieck who defended her thesis on Gr-categories (now called weak 2-groups). The thesis, handwritten and in French, can be found at https://pnp....
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 ...
3 votes
2 answers
337 views

$R$-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\...
5 votes
0 answers
87 views

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 ...
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 ...
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 ...
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 ...
18 votes
3 answers
1k views

Is there a natural measurable structure on the $\sigma$-algebra of a measurable space?

Let $(X, \Sigma)$ denote a measurable space. Is there a non-trivial $\sigma$-algebra $\Sigma^1$ of subsets of $\Sigma$ so that $(\Sigma, \Sigma^1)$ is also a measurable space? Here is one natural ...
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 ...
5 votes
1 answer
495 views

When is a right lifting property closed under pushouts?

A class of morphisms defined by a right Quillen lifting property (weak orthogonality) is always closed under pullbacks (limits); under what assumptions will it be closed under pushouts (colimits)? In ...
7 votes
3 answers
383 views

Yves Diers's thesis ("Catégories localisables")

I am looking for a copy of Yves Diers's 1977 thesis Catégories localisables, which is the original reference for "multi-" category theory, such as multi-adjoints, multi-colimits, and so on. ...
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 ...
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 ...
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 ...
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: $\bullet \to \bullet$, ...
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 ...
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 ...
8 votes
2 answers
1k views

Is there one binary operation foundational for set theory?

The membership relationship "$\in$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\in$". Naturally, the ...
5 votes
1 answer
143 views

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 ...
25 votes
2 answers
760 views

Explicit left adjoint to forgetful functor from Cartesian to symmetric monoidal categories

There is a forgetful functor from the category of (small) Cartesian monoidal categories (a symmetric monoidal category in which the tensor product is given by the categorical product) to the category ...
1 vote
0 answers
81 views

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 ...
28 votes
1 answer
2k views

Analogy between the exterior power and the power set

The symmetric algebra of an object exists in every cocomplete $\otimes$-category. For the category of sets $\mathrm{Sym}(X)$ is the set of multi-subsets of $X$. The usual definition of the exterior ...
1 vote
1 answer
211 views

Pointwise Kan extensions VS weighted limits

$\newcommand{\Dist}{\operatorname{Dist}} \newcommand{\Ran}{\operatorname{Ran}} \newcommand{\Lim}{\operatorname{Lim}}$ TLDR Given a pointwise kan extension, how can we go from $$ \Dist(B, C)(\phi_c \...
15 votes
1 answer
1k views

Gelfand-Naimark from the category-theoretic point of view

I was thinking about the Gelfand-Naimark theorem asserting the isometric * isomorphism between a commutative $C^*$-algebra (with unit) $\mathcal{A}$ and the $C^*$ -algebra of continuous complex-valued ...
9 votes
1 answer
349 views

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 ...
12 votes
2 answers
359 views

Are there finitely many sieves on each object in the distributive lattice cube category?

Define the distributive lattice cube category or Dedekind cube category $\square_{\land\lor}$ to be the full subcategory of the category of posets and monotone maps consisting of objects of the form $[...
13 votes
3 answers
967 views

Model Structure/Homotopy Pushouts in topological monoids?

Let $\mathsf C$ be the category of topological monoids, that is, the category of monoids in $(\textsf{Top}, \times)$. Can the model category structure on $\textsf{Top}$ (Serre fibrations, ...
7 votes
1 answer
310 views

Fibration on the category of Lie pseudoalgebras implementing comorphisms

I am trying to understand comorphisms of Lie pseudoalgebras from the point of view of fibred categories, but failing miserably so far. My question would be: Is there a (op)fibration $\mathrm{LiePs} \...
11 votes
1 answer
578 views

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 views

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 $(V,\rho,\Delta)$ ...
5 votes
1 answer
136 views

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 ...
2 votes
1 answer
126 views

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 ...
7 votes
1 answer
413 views

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 ...
6 votes
1 answer
172 views

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 ...
11 votes
1 answer
528 views

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 views

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
183 views

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
2 answers
219 views

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 ...
1 vote
1 answer
122 views

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
0 answers
69 views

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 $\...