Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,394
questions
4
votes
0
answers
93
views
Monads inside the Kleisli 2-category of another monad
$\require{AMScd}$Let $\cal C$ be a 2-category. Every 2-monad $T$ on $\cal C$ induces a Kleisli 2-category where
0-cells are the objects of $\cal C$
a 1-cell $X\looparrowright Y$ consists of a 1-cell $...
- 13.2k
16
votes
1
answer
491
views
Is there an "injective version" of the Bergner model structure?
The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions.
...
- 61.6k
3
votes
0
answers
117
views
What is the structure required to construct this homotopy of maps between mapping cones?
Let $X,Y$ be topological spaces, let $f$ be a continuous map from $X$ to $Y$ and let $g$ be a continuous map from $Y$ to $X$. Write $C_f$ for the mapping cone of $f$; i.e., $\{*\} + X\times I + Y$, ...
- 897
14
votes
1
answer
357
views
How are MTCs permuted by the Galois action on the little disk operad?
There is a well-studied action of $\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)$ on (some version of) the $E_2$ operad; see for example this MO question.
Modular tensor categories are examples of $...
- 53.1k
8
votes
1
answer
225
views
Class of maps in localized category may not be a set
In one of the very first sentences in Hovey's "Model Categories", Ist chapter, we read that
One can always invert these "weak equivalences"
formally
,
but
there
is a foundational
...
- 1,759
4
votes
0
answers
172
views
Lunatic objects in the 2-category of 2-vector spaces
$\def\duevec{2{\rm Vect}}$See here for the notation.
The 2-category of 2-vector spaces is extremely rich in structure: I'm interested in studying some of its properties in the following perspective.
...
- 13.2k
9
votes
2
answers
804
views
Is the category 2-Vect monoidal closed?
Kapranov and Voevodsky introduced the following 2-category $2{\rm Vect}$ of "2-vector spaces" over a field $K$:
0-cells are natural numbers $\langle N\rangle,\langle M\rangle...$
1-cells $\langle N\...
- 13.2k
5
votes
1
answer
390
views
How much homotopy type theory should be modeled by the unstable motivic category?
It's often said that homotopy type theory should be interpretable in any $\infty$-topos. But really it should be interpretable in any "predicative $\infty$-topos". I'm not quite sure what this means, ...
- 61.6k
1
vote
1
answer
159
views
Abstracting the properties of the category $\frak{g}$-modules
Given a semisimple Lie algebra $\frak{g}$ over $\mathbb{C}$, and a finite dimensional irreducible representation $V$, with dual representation $V^*$, we know that the decomposition of $V \otimes V^*$ ...
- 973
4
votes
0
answers
82
views
Is there a good, general description of morphisms right orthogonal to effective epimorphisms?
Let $C$ be a locally presentable, locally cartesian closed $\infty$-category. Then I think it's not hard to show that the class of effective epimorphisms in $C$ is closed under colimits and cobase-...
- 61.6k
2
votes
1
answer
300
views
On the laplacian of connected, undirected, multigraphs without loops
Let $G$ be a connected, undirected multigraph, without loops.
Let $L_G = D_G - A_G$,
where $D_G= diag (val (v_1), \ldots , val (v_n) )$ where $n$ is the no. of vertices of $G$ and $val (v_i)$ ...
user111492
7
votes
2
answers
916
views
Enrichments vs Internal homs
Consider the definition of existence internal homs for a general monoidal category category $\cal{C}$, mainly the existence of an adjoint for the functor
$$
X \otimes -: \cal{C} \to \cal{C},
$$
for ...
- 973
9
votes
4
answers
989
views
The dual of a dual in a rigid tensor category
For a rigid tensor category $\cal{C}$, can it happen that, for some $X \in {\cal C}$, we have that $X$ is not isomorphic to $(X^{*})^*$, for $*$ denoting dual? If so, what is a good example.
- 973
4
votes
0
answers
139
views
Categorical setting for representations of topological groups in classical sense
Let $G$ be a topological group. A representation of $G$ in a Banach space $V$ is a continuous function $G\times V\to V$.
This MSE question asks about a possible categorical framework for ...
- 10.3k
1
vote
3
answers
328
views
Is there any meaningful extension of the notion of a vector space for multisets?
Consider, as a motivating example, the multiset $\left(\mathbb{P}_n,2\setminus\mathbf{0}\right)$ consisting of the underlying set of polynomials of order $n$ and lower, all with multiplicity $2$—...
5
votes
1
answer
368
views
Homotopy limit of model categories in the category of categories
Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or ...
- 7,902
6
votes
0
answers
127
views
Internal van Kampen colimits
Let $C$ be a category with pullbacks. Recall that a colimit in $C$ is van Kampen if it is preserved by $C/(-): C^{op} \to CAT$, $c \mapsto C/c$.
We can $C$-internalize everything in sight:
Let $\...
- 61.6k
11
votes
1
answer
646
views
The étale topos of a scheme is the classifying topos of which groupoid?
[Sent here from Math.StackExchange by suggestion of an user.]
By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. ...
- 405
3
votes
0
answers
206
views
Categorical features of Hilbert spaces
Does the category of Hilbert spaces and bounded maps have any particular categorical feature which can be studied systematically?
I mean, I know that it's a $*$-category, but it seems to have much ...
user95283
-3
votes
1
answer
3k
views
Why sheaves are important and why do we care about them? [closed]
Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:
$$P:C^{op}\to Set.$$
For every topology $J$ on $C$ we can generate a reflexive subcategory
$$Sh(...
- 369
9
votes
1
answer
461
views
Is every set smaller than a regular cardinal, constructively?
Constructively, my only interest in regular cardinals is in terms of the “$\Sigma$-universes” they generate. By a $\Sigma$-universe, I mean a collection of triples $(X,Y,f: X \to Y)$ closed under base ...
- 61.6k
3
votes
0
answers
163
views
What if adjoints are just too big to give unit and counit?
$\def\CAT{\mathbf{Cat}}\def\Set{Set}\def\bsP{P}$In the category $\CAT$ of possibly large categories, one can build a pseudo-adjunction $P^\sharp \dashv P$ where
$P$ is the functor $A\mapsto [A°,Set]$;...
- 13.2k
2
votes
1
answer
107
views
An immersion of $\text{Sub}(X) \to \text{EqRel}(X)$ in a Malcev category
In an abelian category, each subobject $A \stackrel{f}{\to} X$ individuate an equivalence relation $R(f) \to X^2$ which is given by the equalizer of $$X^2 \rightrightarrows X \to \text{Coker}(f). $$
...
- 8,186
4
votes
0
answers
136
views
Ambidextrous Yoneda structures, admissibles and adjunctions
Let $\cal K$ be a 2-category, supporting two Yoneda structures, induced by a pseudoadjunction $P^\sharp\dashv P : {\cal K}^\text{coop}\to \cal K$; this means that there is such an adjunction, or in ...
- 13.2k
8
votes
2
answers
268
views
Functors in Isbell duality exchange $f^*a$ and $f_*a$
As you maybe remember, Isbell duality is an adjunction
$$\mathcal O : [A°,Set] \leftrightarrows [A,Set]° : {\cal S}pec$$
as defined here; since every functor $f : A\to B$ defines both
a functor $f^...
- 13.2k
8
votes
2
answers
869
views
What is the polynomial functor for the Bag monad
I may be wrong, but we should be able to write the Bag monad in a polynomial form. The bag monad, is exectly the multiset monad whose category of algebras are the commutative monoids. Another name ...
- 1,319
7
votes
1
answer
344
views
Are sifted (2,1)-colimits of fully faithful functors again fully faithful? (And a de-categorified variant)
1) Suppose that I have a sifted diagram of categories $\mathcal{C}_i$, another of the same shape $\mathcal{D}_i$, and that I have a system $F_i:\mathcal{C}_i\to\mathcal{D}_i$ commuting with the ...
- 6,053
6
votes
1
answer
342
views
What is the relationship between $O_G$-parameterized $\infty$-categories and $\infty$-categories enriched in $Top_G$?
Let $G$ be a finite group. Barwick et al define a $G-\infty$-category to be a fibration over the orbit category $O_G$ of transitive $G$-sets. But in the non-$\infty$-land, the natural guess at where I ...
- 61.6k
18
votes
2
answers
1k
views
Category theory for a set/model theorist
I am looking for a book or other reference which develops category theory 'from the ground up' assuming a healthy background in set and model theory, not one in homological algebra or Galois theory ...
28
votes
0
answers
2k
views
Is Feferman's unlimited category theory dead?
In 2013 Solomon Feferman in Foundations of unlimited category theory: what remains to be done (The Review of Symbolic Logic, 6 (2013) pp 6-15, link) laid out three desirable axioms for "...
- 1,436
4
votes
2
answers
324
views
Categorification of spaces and models for set theory
One aspect of topos theory is that it provides an enlarged view of the classical concept of space. Indeed, one may thought that the notion of topos is a sort of categorification of the notion of space....
user95283
6
votes
0
answers
153
views
Extending a functor up to homotopy
Consider a functor between small categories $i:\mathcal{A}\to \mathcal{B}$ which is faithful, not full, and bijective on objects. Consider a functor $F:\mathcal{A}\to \mathcal{M}$ where $\mathcal{M}$ ...
- 3,759
10
votes
2
answers
671
views
On functors preserving monoid objects
If $C$ is a monoidal category, we can define the category $Mon(C)$ of monoids in $C$; call $U_C : Mon(C) \to C$ the forgetful functor. I'm interested in functors between categories of monoids:
...
- 603
2
votes
1
answer
203
views
Left Kan extension and extension of functors
Consider a functor $F:\mathcal{A}\to \mathcal{B}$ between two small categories. Let $\mathcal{K}$ be a locally presentable category. Consider a functor $G:\mathcal{A}\to \mathcal{K}$, there is a ...
- 3,759
3
votes
0
answers
67
views
Calculating the intersection of the saturations of a decreasing sequence of morphisms
I have a sequence $W_0\supset W_1\supset \ldots$ of classes of morphisms in a presentable $(\infty,1)$-category $\mathcal{M}$. I can present this $(\infty,1)$-category as a model category, but I'm ...
- 728
4
votes
0
answers
217
views
The Yoneda embedding induces a monad-like structure?
This question was already asked on MSE.
In CAT the Yoneda embedding $C^{\text{op}} \stackrel{y_C}{\to} \text{hom}(C, \text{Set})$, induces a map $$\text{hom}(\text{hom}(C, \text{Set}), \text{Set}) \...
- 8,186
12
votes
0
answers
411
views
Biased vs unbiased lax monoidal categories
There are two principal ways to define a monoidal category:
The biased definition includes a unit object $I$, a binary tensor product $A\otimes B$, and a ternary associativity isomorphism $(A\otimes ...
- 65.1k
0
votes
0
answers
228
views
Essentially small but not well-powered category?
A category $\mathcal{C}$ is called well-powered if for any $X \in \mathcal{C}$ the class $\mathrm{Sub}(X)$ of subobjects of $X$ is a set. It is called essentially small, if the class of isomorphism ...
user78294
17
votes
2
answers
1k
views
What kind of category is generated by Cubical type theory?
What kind of ‘category’ is Cubical type theory the internal language of?
Its known that Martin-Löf type theories are the internal language of Locally cartesian closed categories, adding higher ...
- 1,185
6
votes
2
answers
852
views
Homotopy for functors
I am reading this paper on Homotopy for functors by Ming-Jung
Lee.
The author gives a definition (at the beginning of section $3$) as follows :
Let $\varphi,\varphi':\Lambda\rightarrow \Gamma$ ...
- 5,931
5
votes
1
answer
123
views
Left split subobject in a $2$-category
Let $\mathcal{K}$ be a $2$-category. Keeping in mind the Cat-intuition on $\mathcal{K}$ I say that:
Def : A $1$-cell $A \stackrel{f}{\to} B$ is a left split subobject if $f$ is a right adjoint and ...
- 8,186
7
votes
2
answers
738
views
Which large cardinals have a Matryoshka characterization?
What on Earth do Russian Matryoshka dolls have in common with large cardinal axioms?! Well, the answer lies in Jónsson algebras! Here is how:
As illustrated in the pictures, a Matryoshka set is a self-...
3
votes
1
answer
236
views
Question on Eilenberg-Watts theorem
I'm not sure if this is a research level question, but:
Let $F:Rep_A \to Rep_B$ be an exact cocomplete functor between representation categories of finite dimensional $k$ algebras, where $k$ has ...
- 1,661
5
votes
1
answer
1k
views
Morita equivalence of Lie groupoids
I am trying to understand what exactly is the Morita equivalence of Lie groupoids.
I am reading Ieke Moerdijk’s notes Orbifolds as groupoids.
A homomorphism $\phi:\mathcal{H}\rightarrow \mathcal{G}$ ...
- 5,931
6
votes
3
answers
389
views
Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom
This question begins with a sort of mysterious comment at the bottom of this Wikipedia page on injective cogenerators. There, it is said, without citation or proof, that as a result of the Tietze ...
- 10.1k
8
votes
1
answer
442
views
Left Bousfield localization without properness, what is known?
I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those ...
- 40.2k
9
votes
2
answers
583
views
2-natural operations on toposes
Any pseudonatural endomorphism $\Phi$ of the forgetful 2-functor $U:Topos^{coop}\to Cat$ is essentially determined by its component $\Phi_{Set}$. But which endofunctors of $Set$ induce such a $\Phi$? ...
- 61.6k
6
votes
3
answers
521
views
Transporting a model category structure along a left adjoint
There is a well-known theorem for transporting a model category structure along a left adjoint $F:\mathcal{M}\to \mathcal{N}$ which is explained here and which is due to Sjoerd Crans.
The difficult ...
- 3,759
1
vote
0
answers
172
views
Modeling scientific theories with category theory (or, how to represent a biological system categorically)
Suppose I wanted to compare Linnaean classification, which arranges species by similarity in ranked taxa, to modern phylogenetic systematics, which appeals to descent with modification and branching ...
- 87
42
votes
3
answers
4k
views
The Origin(s) of Modular and Moduli
In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/...
- 10.3k