Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,364
questions
10
votes
1
answer
658
views
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 ...
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 ...
0
votes
0
answers
91
views
$Q(f+g)_*=Q(f_*+g_*)$ (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 $X$ ...
7
votes
1
answer
208
views
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 ...
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 ...
3
votes
0
answers
123
views
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 ...
12
votes
1
answer
1k
views
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 ...
5
votes
2
answers
821
views
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 ...
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 ...
5
votes
4
answers
970
views
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 ...
2
votes
0
answers
109
views
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 ...
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 ...
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.
...
2
votes
1
answer
155
views
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 ...
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 ...
1
vote
0
answers
113
views
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 ...
8
votes
1
answer
359
views
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 ...
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 ...
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 $G$ ...
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 ...
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 ...
4
votes
2
answers
383
views
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 ...
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 ...
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 $\infty$-...
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 ...
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 ...
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 ...
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}...
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 ...
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 ...
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 ...
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
$$...
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 ...
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 ...
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,...
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 ...
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. ...
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 $\...
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 ...
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 ...
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 ...
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{...
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 ...
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] \...
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-...
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, ...
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}
...
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 ...
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, ...