Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,392
questions
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0
answers
42
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 ${\...
- 12.1k
1
vote
0
answers
118
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?
- 537
2
votes
1
answer
266
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 (...
- 778
5
votes
1
answer
373
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 ...
- 61.5k
4
votes
0
answers
477
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 ...
user139951
9
votes
0
answers
216
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 ...
user30211
2
votes
0
answers
29
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 ...
- 1,816
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0
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196
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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 ...
- 35
0
votes
2
answers
251
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 $f : \Omega \rightarrow X$ ...
- 105
29
votes
0
answers
2k
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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 ...
7
votes
0
answers
290
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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\...
- 1,705
1
vote
1
answer
299
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 ...
user57432
10
votes
0
answers
382
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 ...
- 1,318
1
vote
0
answers
159
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 ...
- 2,890
2
votes
0
answers
58
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 (...
- 171
6
votes
1
answer
246
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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 ...
- 631
34
votes
8
answers
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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 ...
9
votes
3
answers
646
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 $0$) ...
- 2,969
37
votes
4
answers
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 ...
- 21.5k
3
votes
0
answers
70
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
1
answer
175
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 ...
- 363
5
votes
1
answer
182
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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_{...
3
votes
0
answers
96
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 ...
- 26.1k
3
votes
0
answers
103
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 ...
- 1,623
4
votes
2
answers
232
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:
$$Sp \to C \otimes C^{dual}.$$
...
- 1,986
3
votes
1
answer
139
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,...
- 8,559
6
votes
1
answer
183
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 ...
- 26.1k
4
votes
0
answers
152
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 ...
- 4,436
26
votes
1
answer
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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 $X$ ...
- 966
19
votes
4
answers
2k
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?
user138292
6
votes
1
answer
324
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 \...
- 1,623
7
votes
0
answers
168
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 ...
- 437
0
votes
2
answers
588
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 ...
user137767
7
votes
1
answer
396
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 ...
- 111
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votes
0
answers
179
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....
- 646
6
votes
1
answer
203
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 ...
- 295
9
votes
2
answers
980
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 ...
- 332
6
votes
0
answers
380
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 ...
- 718
1
vote
1
answer
207
views
Product of two group morphisms not a group morphism
In Mac Lane's Categories for the Working Mathematician, page 110, second edition, he states that, in the category of groups $Grp$, $F_n$ being groups and $x_n, y_n$ being two cones on these groups (...
- 507
3
votes
1
answer
243
views
Category of continuous self maps
Is there any way to reconstruct a topological space from the category of its continuous self maps (possibly under some assumptions)?
How can we tell whether a category is the category of continuous ...
- 2,582
4
votes
1
answer
323
views
Thomason fibrant replacement and nerve of a localization
The Thomason model structure on the category $\mathrm{Cat}$ of small categories is transferred along the right adjoint of the adjunction $$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \...
- 1,223
12
votes
3
answers
696
views
On model categories where every object is bifibrant
Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have various general constructions that allow (under some assumption) to go from one ...
- 40.2k
6
votes
1
answer
396
views
Isomorphisms in enriched categories
Let $(M,\otimes,1)$ be closed monoidal category and $C$ an $M$-enriched category. Assume we have $C$-objects $X$ and $X'$ and a morphism $f:1\to C(X,X')$ in $M$. We call $f$ an isomorphism if there is ...
- 1,623
18
votes
2
answers
731
views
Properties of categories that can not be proven by abstract nonsense
What are examples of properties of particular categories that can be formulated in categorical language and "feel" like they ought to be provable formally but they actually are not? I think that this ...
user137597
6
votes
1
answer
194
views
Two monoidal structures and copowering
Let $(\mathbf{M},\otimes,1)$ be a closed monoidal category and $(\mathbf{C},\oplus,0)$ an $\mathbf{M}$-enriched monoidal category. Furthermore, assume that we have a copowering $\odot:\mathbf{M}\times\...
- 1,623
7
votes
2
answers
264
views
The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$
Very often, in topology, one restricts to a coreflective Cartesian closed subcategory of $\mathbf{Top}$ in order to freely use exponential laws for mapping spaces, which imply things like "the ...
- 6,661
4
votes
1
answer
380
views
Idempotent completion of linear categories and Yoneda
Let $ \text{Vect} $ be the category of finite dimensional vector spaces over an algebraically closed field. The idempotent completion of a $\text{Vect}$-category $ \mathcal{C} $ may be though of in ...
- 1,379
4
votes
0
answers
288
views
EGA I (Springer), Proposition 0.4.5.4 [closed]
I do not understand one argument in the proof of Proposition 0.4.5.4. in the new version by Springer of EGA I.
When proving that the functor $F$ is representable by $(X, \xi)$, where we obtained $X$ ...
- 365
14
votes
2
answers
340
views
Example of non accessible model categories
By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible in this sense (and just in case: even by using Vopěnka's ...
- 3,759
7
votes
0
answers
95
views
Terminology: a certain semicategory with objects mor(C) (not the usual or twisted arrow category)
I’ve had recent cause to consider the following construction: given a category $\newcommand{\C}{\mathbf{C}}\C$, define a semicategory $M(\C)$, whose objects are arrows of $\C$, and where a map from $f ...
