Questions tagged [ct.category-theory]

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

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On HTT's Lemma 3.3.4.1

While studying the book Higher Topos Theory I have encountered some difficulty with Lemma 3.3.4.1, which says that the pullback along a cartesian fibration of a map q such that $q^{op}$ is cofinal is ...
Edoardo Lanari's user avatar
7 votes
2 answers
594 views

What is the name for a set endowed with a Lipschitz structure?

I am interested in the standard (or widely accepted) name for a mathematical structure, which is intermediate between the structures of a metric space and a topological space. I have in mind the ...
Taras Banakh's user avatar
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$Q(f+g)_*=Q(f_*+g_*)$ (The maps induced by the sum is the sum of induced maps modulo decomposables [Reference request]

Let $X, Y$, let's say, homotopy commutative $H$-spaces, $f,g$ maps from $X$ to $Y$. (Actually we only need $Y$ to be homotopy commutative $H$_space, but the statement is easier if we also suppose $X$ ...
user43326's user avatar
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7 votes
1 answer
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Saturated classes, generation by a set and pullbacks of categories

Assume that we have a pullback square $$ \begin{array}{ccc} A & \rightarrow & B \\ \downarrow & & \downarrow \\ C & \rightarrow & D \\ \end{array} $$ with all functors ...
Edouard's user avatar
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2 votes
1 answer
497 views

What is the (Co)Monad for a Bag

A Bag is a data structure, like a list, that stores items with no concept of order. The only operations on the structure is to add an item and then iterate through the items with no guarantee as to ...
Ben Sprott's user avatar
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3 votes
0 answers
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Additional structure on augmented simplicial sets

I am currently studying augmented simplicial sets with some additional degeneracies. I was wondering if it is a structure that was already identified somewhere. This additional data is a family of ...
Maxime Lucas's user avatar
12 votes
1 answer
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Is the derived category of $A$-dg-modules as a dg-category coincide with the ordinary definition of derived category?

Let $A$ be a unital dg-algebra over a base field $k$. We consider the category of (unbounded) right $A$-dg-modules with morphisms closed degree $0$ maps. We denote this category by dg-mod-$A$. We ...
Zhaoting Wei's user avatar
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5 votes
2 answers
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Can a nontrivial presheaf have a trivial sheafification?

I'm studying Vakil's Foundations of Algebraic Geometry, and working through the exercises in 2.4 about compatible germs as a method for constructing a sheafification of a given presheaf. Can this ...
Jeremy Gross's user avatar
32 votes
1 answer
2k views

What was the error in the proof of Roos' theorem?

Background: In 1961, Roos (who, sadly, apparently passed away just last month) purported to prove [1] that in an abelian category with exact countable products (AB4${}^\ast_\omega$), limits of inverse ...
Tim Campion's user avatar
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4 answers
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What are the adjunctions that generate the Giry Monad?

The Giry Monad captures probability measures. What is the adjunction that generates the Giry Monad? To narrow this down, perhaps we can talk about the adjunction between the category of Polish ...
Ben Sprott's user avatar
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2 votes
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Commutativity of simple diagrams built out of canonical natural transformations

Let be a square of right adjoints which is commutative up-to a natural isomorphism $\varphi\colon v_*f_* \to g_*u_*$ (one can suppose it is the identity), where the left adjoint of $f_*$ is denoted ...
Andrea Gagna's user avatar
12 votes
1 answer
519 views

Is there a "killing" lemma for G-crossed braided fusion categories?

Edit: I found a serious flaw in the question and my answer, and I had to change a lot. The basic question is still there, but the details are a lot different. Premodular categories In braided ...
Manuel Bärenz's user avatar
5 votes
0 answers
180 views

Kan extensions and orthogonality

$\require{AMScd}$I have the feeling that there is a connection between the existence of a Kan extension in a 2-category ($\bf Cat$ will be sufficient) and a 2-dimensional notion of orthogonality. ...
fosco's user avatar
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2 votes
1 answer
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A non-monoidal functor that respects fusion rules

Let $(\cal{C},\otimes)$ and $(\cal{D},\odot)$ be two monoidal categories. Moreover, assume that $\cal{C}$ and $\cal{D}$ are abelian and semisimple. Let $X,Y$ be two simple objects in $\cal{C}$, and ...
Abo Kutis-Felan's user avatar
14 votes
2 answers
652 views

Status of Thomason's idea for a symmetric monoidal model of stable homotopy - from his last paper

In 1995, Robert Thomason published “Symmetric monoidal categories model all connective spectra” in TAC. On page 2, he argues that symmetric monoidal categories are more convenient than “May’s ...
David White's user avatar
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1 vote
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Selection in a small category

I came across the following problem. I do not know if these notions are known (I would actually be interested to know), so the names might not be the canonical ones. Given a small category $C$, a ...
Jeremy's user avatar
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1 answer
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Is every category equivalent to the fundamental category of a directed space?

I was wondering if there was some work in directed algebraic topology to define a classifying directed space of a category? By that I mean the following. In (undirected) algebraic topology, we ...
Jeremy's user avatar
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3 votes
2 answers
806 views

Coverings of a space and coverings of a groupoid

In algebraic topology, there is the well-known notion of coverings of a space. It is very nice, it has many properties, but I find it frustrating that: 1) some hypotheses are needed for them to work ...
Jeremy's user avatar
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7 votes
0 answers
216 views

$\text{Log}_2$ of set functors

Given a functor $T: \text{Set} \to \text{Set}$ is there a way to understand if it is of the form $2^G$ for a functor $G: \text{Set} \to \text{Set}$? I am looking for a condition like if such a $G$ ...
Ivan Di Liberti's user avatar
8 votes
1 answer
288 views

How are the left and the right group of a bitorsor related?

This question arose from my answer to To what extent does a torsor determine a group: it turns out that I do not know one thing about it. Let $G$, $G'$ be groups in some nice enough category (you may ...
მამუკა ჯიბლაძე's user avatar
11 votes
2 answers
1k views

Is every "nice" abelian category with enough projectives an additive presheaf category?

A "nice" category $\mathcal{C}$ should be (for the purposes of this question) locally presentable at a minimum, and maybe a bit more. One might require $\mathcal{C}$ to be (in roughly order of ...
Tim Campion's user avatar
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4 votes
2 answers
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Ge-categories and applications

Ge-categories, i.e., categores enriched over groupoids (these are 2-categories where the set of morhisms $HOM(a,b)$ has a groupoid structure) seem to be useful in homotopy theory. Question: What are ...
user avatar
8 votes
0 answers
192 views

Day convolution for prederivators

Let $\cal J = \bf Cat$ be the strict 2-category of small categories, functors and natural transformations and $\mathbb{V} : {\cal J}°\to \bf MonCAT$ a strict 2-functor taking value on possibly large ...
fosco's user avatar
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9 votes
2 answers
625 views

If the homotopy category is well-generated, must the $\infty$-category be presentable?

Suppose $\mathcal{C}$ is a stable $\infty$-category whose homotopy category is a well-generated triangulated category in the sense of Neeman's book. Must $\mathcal{C}$ be a presentable $\infty$-...
David White's user avatar
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11 votes
1 answer
803 views

Colimits, limits, and mapping spaces

It is true that in the category of topological spaces $ \mathrm{Map}(\underset{i\in I}{\mathrm{colim}}\, X_i, Y)\cong \underset{i\in I}{\mathrm{lim}}\,\mathrm{Map}(X_i,Y)$ ? Here mapping spaces are ...
Victor's user avatar
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21 votes
2 answers
445 views

Is the order on repeated exponentiation the Dyck order?

The Catalan numbers $C_n$ count both the Dyck paths of length $2n$, and the ways to associate $n$ repeated applications of a binary operation. We call the latter magma expressions; we will ...
David Spivak's user avatar
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2 votes
1 answer
1k views

Projectives in the category of quasi coherent sheaves Qch(X)

X is a non singular projective variety over an infinite field k. How to prove there are no projectives with a surjective map to the structure sheaf O_X in Qch(X) and Coh(X). Coh(X) is the category of ...
Rick Sanchez's user avatar
4 votes
1 answer
836 views

Dual objects in the $\infty$-category of spectra

We say (according to https://ncatlab.org/nlab/show/%28infinity%2Cn%29-category+with+duals) that a symmetric monoidal $(\infty,1)$ category $\mathcal{C}$ has duals if its homotopy category $h\mathcal{C}...
Exit path's user avatar
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11 votes
3 answers
2k views

What is the symmetric monoidal structure on the $(\infty,1)$-category of spectra?

The $(\infty, 1)$ category $Sp$ of spectra as defined by Lurie in Higher Algebra has the structure of a symmetric monoidal category. Although I know the definition of symmetric monoidal category in ...
Exit path's user avatar
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25 votes
1 answer
2k views

Locally presentable abelian categories with enough injective objects

I came to the following question when thinking about the (infinitely generated) tilting-cotilting correspondence, where it appears to be relevant. Does there exist a locally presentable abelian ...
Leonid Positselski's user avatar
51 votes
8 answers
14k views

The "derived drift" is pretty unsatisfying and dangerous to category theory (or at least, to me) [closed]

I'm currently a young, not-so-young mathematician, finishing its second postdoc. I developed an interest for rather different topics in the last few years but constantly, slowly converged towards ...
3 votes
1 answer
198 views

Co/completeness of truncated 2-category

There seems to be many ways to obtain a 1-category out of a 2-category: Dumb truncation. $\delta: 2\text{-Cat} \to \text{Cat}$ sends a 2-category $\cal K$ into the 1-category obtained forgetting the ...
fosco's user avatar
  • 13k
6 votes
1 answer
345 views

Complexes in stable categories

Generalizing from 1-category theory, there's a simple definition of a "naive complex" in a stable $\infty$-category. Considering bounded positive graded chain complexes, they are a sequence of maps $$...
user avatar
2 votes
0 answers
122 views

Pro-discrete modules vs modules over the completed ring

Suppose that $R$ is a topological ring with a complete system of neighbourhoods of zero given by left ideals. Let $\widehat{R}$ be its completion. Are the following four categories equivalent (perhaps ...
Balerion_the_black's user avatar
9 votes
2 answers
3k views

Classifying space as the geometric realization of the nerve of $G$ viewed as a small category

Let $C$ be a small category: we define its nerve $(N(C)_k)_k$ as the following simplicial set: $N(C)_0=Ob(C)$ (the set of objects), $N(C)_1=Mor(C)$ (the set of all morphisms) and $N(C)_k$ to be a set ...
truebaran's user avatar
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3 votes
1 answer
225 views

Accessible categories in enriched category theory

I study some definitions of accessible category (see 1) and the applications of that notions; my question: exist a notion of accessible category in therm of enriched category theory? (in case of exist,...
Wilson Forero's user avatar
9 votes
2 answers
362 views

How to compute (enriched) Cauchy completions?

Lawvere famously explained that the following three constructions are all secretly "the same" construction: Completing an ordinary category by including splittings of all idempotents. Completing a ...
Theo Johnson-Freyd's user avatar
7 votes
1 answer
243 views

"Monoid objects" without points

Let $\mathbf{C}$ be a category with binary Cartesian products. Let's say that a map $f : X \to Y$ is constant if, for every pair of maps $g,h : A \to X$, maps $f \circ g$ and $f \circ h$ are equal. ...
Valery Isaev's user avatar
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3 votes
1 answer
301 views

What is the name for a natural transformation that has both lax and oplax monoidal properties?

Let $\mathcal C,\mathcal D,\mathcal E$ be monoidal categories, let $g$ be an oplax monoidal functor from $\mathcal C$ to $\mathcal D$ and let $G$ be a lax monoidal functor from $\mathcal D$ to $\...
John Gowers's user avatar
8 votes
2 answers
456 views

"Equivalence" is to "group" as "adjoint" is to ....?

The collection of all self-equivalences of a category $C$ constitutes a $2$-group, which is a categorification of the notion of a group. My question is about what happens when one replaces ...
user85913's user avatar
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4 votes
0 answers
109 views

Reference to an explixit construction of a locale from a measurable space

In A sheaf theoretic approach to measure theory shows that measures on a measurable space are equivalent to measures on some locale whose open sets are the $\sigma$-ideals of the $\sigma$-algebra. The ...
Lolman's user avatar
  • 369
5 votes
1 answer
273 views

Factorization of Gabriel-Zisman localization construction?

My question concerns whether the Gabriel-Zisman localization construction $S^{-1}$ for categories admits a known factorization into a pair of commuting constructions $S^l$ and $S^r$. The localization ...
Ben Cooper's user avatar
10 votes
1 answer
658 views

Persistent homology over the integers

Is it likely that in the future, there will be interest in computing persistent homology over the integers (or other PIDs)? Currently, persistent homology is usually done over a field (like $\mathbb{...
yoyostein's user avatar
  • 1,219
13 votes
1 answer
491 views

Functorial multiplication on commutative rings

Let $C$ be the category of associative commutative rings with 1 and let $F:C\to C$ be a functor which commutes with the forgetful functor to abelian groups (i.e. $F$ is a functorial way to define ...
Alexander Braverman's user avatar
14 votes
2 answers
519 views

Constructive proofs of existence in analysis using locales

There are several basic theorems in analysis asserting the existence of a point in some space such as the following results: The intermediate value theorem: for every continuous function $f : [0,1] \...
Valery Isaev's user avatar
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7 votes
1 answer
212 views

Characterizing freely adjoining K-filtered colimits as K-continuous presheaves

Note: This question has a 1-categorical and an $\infty$-categorical versions. I am interested in the $\infty$-categorical one so this is the version that I write below, but an answer for the 1-...
KotelKanim's user avatar
  • 2,270
11 votes
1 answer
2k views

What are compact objects in the category of topological spaces?

Let $\mathscr C$ be a locally small category that has filtered colimits. Then an object $X$ in $\mathscr C$ is compact if $\operatorname{Hom}(X,-)$ commutes with filtered colimits. On the other hand, ...
R. van Dobben de Bruyn's user avatar
2 votes
0 answers
140 views

What name can I use for a cocone over the mapping cone diagram?

In homotopy theory, the mapping cone of a continuous map $f\colon X \to Y$ is the homotopy pushout over the following span: $$ \require{AMScd} \begin{CD} X @>{f}>> Y\\ @VVV \\ \{*\} \end{CD} ...
John Gowers's user avatar
15 votes
0 answers
520 views

Is this an $E_\infty$-algebra?

I have a particular kind of algebraic structure that's come up in my work. It's basically a chain complex equipped with a multiplication which is commutative and associative up to homotopy in a ...
Dan Petersen's user avatar
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12 votes
2 answers
1k views

$K$-theory backwards

Let $R$ be a ring. The $K$-theory of $(Mod(R)^{f.g.proj},\oplus)$ is obtained by first throwing out non-isomorphisms and then group completing. What happens if these steps are reversed? That is, ...
Tim Campion's user avatar
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