Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [ct.category-theory]

Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

16
votes
0answers
331 views

Caramello's theory: applications

In this text, the author says (well, he says it in French, but I am too lazy to fix all the accents, so here is a Google translation): In any case, contemporary mathematics provides an example of ...
4
votes
1answer
105 views

Under what conditions is a symmetric tensor category equivalent to $\operatorname{\mathsf{Rep}}G$ for some group $G$?

Deligne's theorem on tensor categories states that for any symmetric tensor category $\mathcal{C}$ satisfying the subexponential growth condition, there is a fiber functor to $\mathsf{sVec}$ and that $...
2
votes
0answers
73 views

Which set of compact objects generates the subcategory of a compactly generated stable model category?

I couldn't find any info on what set of compact objects generates the following subcategory: Let $k$ be a field of positive characteristic and let $G$ be either a finite group or a finite group ...
9
votes
1answer
184 views

Is $\operatorname{Hom}(F,G)$ finite if $F$ and $G$ are endofunctors of the category of finite sets?

I asked this question on Mathematics Stackexchange but got no answer. Are there endofunctors $F$ and $G$ of the category of finite sets such that there are infinitely many natural transformations ...
17
votes
2answers
919 views

How much Replacement does this axiom provide?

(There have been many questions on MathOverflow about the axiom scheme of replacement, including a few with a similar flavour to mine. Some have very informative answers and link to excellent papers ...
5
votes
1answer
895 views

Anabelian geometry ~ higher category theory

Note: I'm worried this question might be taken as controversial, because it relates to Shinichi Mochizuki’s work on the abc conjecture. However, my question has nothing to do with the correctness of ...
4
votes
0answers
181 views

A compendium of weak factorization systems on $sSet$

A (weak) factorization system on a category $\mathcal{C}$ consists of a pair of classes of morphisms in the category $(L,R)$ satisfying Every morphism $f:x \to y \in \mathcal{C}$ can be factored (...
11
votes
2answers
340 views

Extending Kan fibrations, without using minimal fibrations

$\require{AMScd}$One thing that needs to be checked to give an interpretation of type theory in simplicial sets (as in Kapulkin-Lumsdaine) is that "the base of the universal fibration is fibrant". ...
2
votes
0answers
66 views

Definition of gluing of dg categories

I am reading the paper by Kuznetsov and Lunts, Categorical resolutions of irrational singularities, and I’m struggling with a few things. The definition of gluing of DG-categories $\mathcal{D}_1$ and $...
8
votes
0answers
126 views

Understanding a monad from its fixed points

Let $(T, \eta, \mu)$ be monad over $\mathsf{C}$. And let $\iota : \mathsf{Fix}(T) \hookrightarrow \mathsf{C}$ be the inclusion of the full subcategory of fixed points of $T$. By the universal ...
6
votes
0answers
38 views

Special monomorphism to encode the inclusion of topological submonoids

Consider the category $\mathrm{TopMon}$ of topological monoids and continuous monoid homomorphisms. Consider the inclusion $i:\Bbb{R}_{\ge 0}\hookrightarrow \Bbb{R}$, where the spaces are taken with ...
2
votes
0answers
26 views

Preradicals generating a radical

Let ${\mathcal A}$ be a locally Noetherian Grothendieck category. For simplicity you may restrict to the category of R-modules for an associative ring. Subject to knowing the Zigeler spectrum of ${\...
1
vote
0answers
59 views

A construction for the free $\omega$-category generated by a globular set

The forgetful functor from strict $\omega$-categories to globular sets has a left adjoint. Where can one find an explicit construction for this free functor?
3
votes
1answer
204 views

Simplicial set represented by an (unordered) set

Let $X$ be a (finite if you want) set and form the simplicial set $F^{\bullet}(X)$ with $$ F^{n}(X) = \mathrm{Hom}_{\mathrm{set}} ([n], X) $$ where the right hand side denotes arbitrary maps of sets (...
5
votes
1answer
258 views

Are there universal homological functors?

There is a bifunctor $H: Stab^{op} \times Ab \to Top$ where $H(C,A)$ is the space of homological functors $C \to A$. Is this bifunctor left or right representable? That is, for each small abelian ...
0
votes
0answers
66 views

Presentation of enriched categories

For an ordinary category, it is clear to me what a representation is: We have a notion of: Free category over a quiver, Congruence relations (a family of equivalence relations on each $C(x,y)$ such ...
6
votes
1answer
245 views

A slightly canonical way to associate a scheme to a Noetherian spectral space

Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...
7
votes
0answers
107 views

Point-free topology, but with $\sigma$-algebras instead of spaces

I have a question about $\sigma$-algebras in relation to point-free topology. The question was inspired by a comment on a similar question I had: If abstract $\sigma$-algebras (i.e. certain boolean ...
2
votes
0answers
21 views

When do projection maps of polyhedra factor?

Given three polyhedra $P$, $Q$, and $R$ in dimensions $a$, $b$, and $c$ respectively, with $a\leq b\leq c$, with the additional condition that: $P=\pi_1(Q)=\pi_2(R)$, where $\pi_1$ and $\pi_2$ are ...
2
votes
0answers
173 views

Why each functor defines an invariant, but not every invariant is functorial ? Examples? [closed]

In Category Theory each functor defines an invariant, but not every invariant is functorial Why ? Can you provide some examples when a functor is an invariant a invariant is a functorial a ...
0
votes
2answers
182 views

Proving that preorder on the set of measurable functions is symmetric

Let's say I have specific preorder $\prec$ on set $S$ and I want to prove that in fact it is equivalence relation. What is known already: $S$ is set of measurable functions $f : \Omega \rightarrow X$ ...
24
votes
0answers
1k views

Did Grothendieck overestimate topoi?

I was reading the Russian translation of Recoltes et Semailles and in the footnote where Grothendieck lists his 12 contributions (including schemes) we find the following lines: Из этих тем ...
5
votes
0answers
115 views

Tensor-hom adjunction in a general closed monoidal category

Let $(C,\otimes,1)$ be a closed (not necessarily symmetric) monoidal category with all finite limits and colimits and with the internal hom functor $[b,-]$ right adjoint to $(-)\otimes b$, for any $b\...
1
vote
1answer
210 views

Topological Invariants for Group

Let $\mathbf{Grp}$ be the category of groups and $\mathbf{Top}$ be the category of topological spaces. To each group $(G, \circ_G)$, we can associate a topological space $(G,\tau_G)$ the basis for ...
9
votes
0answers
175 views

The term “absolute geometry”

My question concerns the so-called absolute geometry over the "field with one element" F_1 or over the spectrum $\mathrm{Spec}(F_1)$, cf. https://ncatlab.org/nlab/show/Borger%27s+absolute+geometry. I ...
1
vote
0answers
122 views

Equivalence of categories and homotopy equivalence

There is some conventional wisdom that an equivalence of categories is akin to a homotopy equivalence between topological spaces. If I were forced to explain this wisdom, I'd fail miserably, but ...
2
votes
0answers
41 views

direct limit in locally convex modules and continuous map

Let we have short exact sequences of LCM over LC algebra $A$ with continuous linear maps $$ 0\to B_j\;{\xrightarrow {\ f_j\ }}\;C_j\;{\xrightarrow {\ g_j\ }}\;D_j\to 0. $$ We can take inductive limit (...
5
votes
1answer
182 views

Discernible Objects in a Topos

Perhaps an overly elementary question: let $\mathcal{E}$ be a topos and let $X, Y$ be non-isomorphic objects in $\mathcal{E}$. Is it always true that there exists a formula $\phi$ of $\mathcal{E}$'s ...
9
votes
2answers
259 views

Big list of comonads

The concept of a monad is very well established, and there are very many examples of monads pertaining almost all areas of mathematics. The dual concept, a comonad, is less popular. What are ...
8
votes
3answers
422 views

Tannaka duality for semisimple groups

Tannakian formalism tells us that for any rigid, symmetric monoidal, semisimple category $\mathcal{C}$ equipped with a fiber functor $F: \mathcal{C} \to Vect_k$ for a field $k$ (of characteristic $0$) ...
33
votes
4answers
2k views

How many morphisms from 1 to 1+1 can there be?

Here is an interesting question raised by Alice Rhyl. Let $C$ be a category with a terminal object $1$ and finite coproducts. How many different morphisms $f : 1 \to 1 + 1$ can there be? There are ...
3
votes
0answers
48 views

Is there a semisimple Hopf algebra Grothendieck equivalent to a strictly weak one?

By Corollary 2.22 in On fusion categories (by Pavel Etingof, Dmitri Nikshych and Viktor Ostrik) any fusion category is equivalent to the category of finite dimensional representations of a semisimple ...
4
votes
1answer
99 views

Why is the category of all small $\mathbf{S}$-enriched categories locally presentable?

In Lurie's Higher Topos Theory Proposition A.3.2.4, the author used Proposition A.2.6.15 to prove that for any combinatorial monoidal model category $\mathbf{S}$ with all objects cofibrant and weak ...
4
votes
1answer
85 views

Fusion category and induction matrix to its Drinfeld center: combinatorial properties

This question is inspired by this paper of Scott Morrison and Kevin Walker. Consider a fusion category $\mathcal{C}$ of rank $r$, and its Drinfeld center $Z(\mathcal{C})$ of rank $s$. Let $N_i = (n_{...
2
votes
0answers
74 views

When is the category of complexes of finite type?

For a ring $R$ define the category of complexes of length $n \geq 2$ as the category $C_n(R)$ with objects the complexes of the form $0 \rightarrow X_n \rightarrow \cdots X_1 \rightarrow 0$ with the ...
3
votes
0answers
48 views

Generating an enriched multicategory

Let $C$ be an $(M,\otimes,1)$-enriched category. I am looking for a reference for a notion of “generating the morphisms of $C$” (for ordinary categories, but also for multicategories, see below). My ...
3
votes
0answers
72 views

Counit map for compactly generated categories

Any compactly generated presentable stable $\infty$-category $C$ is known to be dualizable (with respect to Lurie's tensor product), so there is a coevaluation map: $$Sp \to C \otimes C^{dual}.$$ ...
3
votes
1answer
126 views

Lawvere metrics on the poset of subgroups of Z?

Background: Recall that a Lawvere metric structure on a set $X$ consists of a function $d\colon X\times X\to[0,\infty]$ satisfying two properties: $d(x,x)=0$ for all $x\in X$, $d(x,y)+d(y,z)\geq d(x,...
6
votes
1answer
119 views

Categories with every indecomposable object being uniserial

Let $A$ be an abelian category. A Jordan-Hölder series for an object $X$ is a filtration $0<X_0<X_1<\cdots<X_n=X$ such that $X_i/X_{i-1}$ are simple. Call $X$ uniserial in case it has a ...
4
votes
0answers
97 views

Image of $\rm{lim}^1$ functor

In category of abelian groups, we know that — values of $\rm{lim}^1$ on countable systems are precisely cotorsion groups — values of $\rm{lim}^1$ on systems of finitely generated groups are of the ...
24
votes
1answer
1k views

Have the Quantum Group Theorists taught the Group Theorists Anything?

I will start with the general before moving to the specific. Consider for a moment the two (very) soft definitions. An abstraction of an object $X$ is a category $\mathcal{C}_0$ such that $X$ ...
11
votes
3answers
663 views

Name for abelian category in which every short exact sequence splits

What is the name of the class of abelian categories defined by the following property: every short exact sequence splits?
6
votes
1answer
242 views

Free operad over a monoid object

Let $\mathcal{O}$ be an operad in the monoidal category $M$. Then $\mathcal{O}(1)$ together with the morphisms $$\mathcal{O}(1)\otimes \mathcal{O}(1)\to \mathcal{O}(1)$$ and the unit $\eta:1\to \...
6
votes
0answers
108 views

How to translate connection on four graphs to quantum 6j symbols

I need the explicit quantum 6j symbols for the Haagerup fusion category for a physics research project. This paper math/9803044 by Asaeda and Haagerup brute-force constructs the Haagerup subfactor, by ...
0
votes
2answers
314 views

Doing scheme theory with Hausdorff spaces

Suppose I have an allergy to non-Hausdorff spaces but I really want to do, say, arithmetic geometry. I wonder if there is some perverse way I could develop scheme theory that would accomodate for my ...
5
votes
1answer
129 views

Beck-Chevalley condition on pushouts

Let $C$ be a regular category with pushouts and $S(X)$ is the lattice of subjects of $X$. For every arrow $f\colon X\to A$, pulling back along $f$ gives a map $f^*\colon S(A)\to S(X)$ which has a left ...
6
votes
0answers
112 views

Abelianization derivator

About ten-fifteen years ago, when the theory of abstract triangulated categories reached a culminating point (after the publication of Neeman's book http://hopf.math.purdue.edu/Neeman/triangulatedcats....
6
votes
1answer
157 views

Non-trivial factor splitting from vacuum in TQFTs. F-move not unity?

Consider a unitary modular TQFT, defined by the F and R moves. More specifically, a braided tensor category relevant for anyon models in 2D topologically ordered phases of matter. I am interested in ...
9
votes
2answers
549 views

A category-like structure without composition?

Is there a name for the 'category-like' structure which satisfies the axioms for a category except for composition, i.e. identities exist for every object, if $f\in Hom(A,B)$ and $g \in Hom(B,C)$ then ...
4
votes
0answers
152 views

When do Kan extensions preserve colimits?

Assume that we have a pair of functors $Y:A \to B$ and $F:A \to C$ where $A$ is an essentially small category, $B,C$ are cocomplete categories and $Y,F$ preserve colimits. Assume also that for some ...