Questions tagged [ct.category-theory]
Enriched categories, topoi, abelian categories, monoidal categories, homological algebra.
6,364
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Reference request for (co-)free constructions
Following a comment of user131781, posted to an answer of this question on MO, I am looking for references to the construction of (co)-free functors from categories into the category of Banach spaces ...
8
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3
answers
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Meaning of A-infinity relations
I am learning A-infinity category with Fukaya category in mind, and would like to understand the meaning of A-infinity relations.
In particular, as $N=1$, it means $dd=0$. As $N=2$, it means that $d$ ...
3
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0
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168
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Connectedness for stacks
Let $X$ be a stack for the Zariski (or etale) site over an arbitrary field $k$. The functor $\pi_0(X) : Alg_k \to Sets$ of path-components of $X$ is defined as the composition
$$Alg_k \overset{X}{\...
11
votes
1
answer
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Slicing up monads on categories with pullbacks: what are these mysterious "zerosumfree" monads"
Introduction
I'll describe a way of taking a monad on a category $\mathcal{E}$ with pullbacks, and obtaining a monad on each slice category. I'll show that this construction is always lax-natural in $...
3
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0
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246
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Pushforward of covering maps
Let $A,B$ be sufficiently connected spaces so as to have the category of coverings equivalent to the category of representations of the fundamental groupoid.
Given a covering map of $A$ and a ...
4
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0
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97
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Partially fibered categories vs T-Multicategories
Short version: This is a reference request question. I would like to know if something has been written on the connection between $T$-multicategory (for $T$ a monad on a category $\mathcal{E}$), and ...
1
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0
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76
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Distributive laws of strong and/or monoidal monads
It is well-known that a commutative strong monad is the same as a monoidal monad.
Is there a notion of distributive law for commutative strong monads which is equivalently a distributive law for ...
7
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0
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Is it a coincidence that $Gal(\mathbb C / \mathbb R) \cong C_2 \cong Aut(E_1)$? (Or: why are $\mathbb C$-algebras with involution so useful?)
The automorphism group $Aut(E_1)$ of the $E_1$ operad is the cyclic group of order 2, $C_2$, and thus $C_2$ acts on any category of algebras (by reversing the multiplication). The seeming coincidence ...
6
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2
answers
313
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Uniqueness of "Limit" of Cyclic Binary Strings
Set-up: By abuse, let $\sigma$ represent both the left shift operator on infinite bi-infinite strings and the cyclic left shift operator on finite strings. (Thus, for example, $\sigma(...01\bar{0}10......
10
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1
answer
599
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Reference request about “internal language of categories”
I've tried to become familiar with the so-called "internal language of a category" for the last months. However, I'm still not confident enough when, for instance, I find a subobject (of a given ...
3
votes
1
answer
614
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Criteria for representability of a functor from schemes to sets
I have read in several texts that in order to show that a given functor from the category of schemes (over, say an algebraically closed field $k$) to $\mathsf{Set}$ is representable, it suffices to ...
14
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1
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599
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Are locally presentable categories determined by their objects?
Let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a colimit-preserving functor between locally presentable categories. Assume that $f$ induces an equivalence between the groupoids underlying $\mathcal{C}...
7
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0
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Example of a tensor triangulated category with two different monoidal t-structures?
What's an example of a tensor triangulated category / symmetric monoidal stable $\infty$-category with two different monoidal $t$-structures?
While I'm at it: is there an example of a tensor ...
4
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0
answers
139
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Adjoint actions in abstract tensor categories
Say we have a Lie group $G$. The category of (finite, complex) representations $\mathsf{Rep}\,G$ contains the adjoint representation $\mathfrak{g}$ which has many special properties. For instance $\...
6
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1
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Is it possible to use Feynman diagrams to represent a dot product $a \cdot b$?
Feynman diagrams are topological entities, but they describe linear
operators
It has been observed that Feynman diagrams are in particular string diagrams (morphisms in monoidal categories)) in ...
18
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1
answer
859
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What are the monomorphisms of ($\infty$-)toposes?
There are standard notions of "surjections" and "embeddings" of toposes. However, not every surjection is an epimorphism, and not every regular monomorphism is an embedding. (EDIT: as Alexander ...
6
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0
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What are the effective epimorphisms of presentable $\infty$-categories?
Let $\mathcal C$ be a sufficiently nice $\infty$-category, and let $f: U \to X$ be a morphism in $\mathcal C$. Recall that $f$ is said to be an effective epimorphism if the induced map $|U^{\times_X (\...
4
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0
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Is well-pointedness the reason that the internal/external distinction seems not to apply to $\mathbf{Set}$?
When reasoning about the category of sets, we usually don't have to worry about the internal/external distinction. For example, if $f : X \rightarrow Y$ is a morphism of sets, then $f$ is either ...
8
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1
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336
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What is the categorical analogue of openness?
Let us say that a category $\mathcal C$ has enough of some class $\mathcal U$ of object if every object in $\mathcal C$ is a colimit of objects of the class $\mathcal U$. The pointset topology ...
0
votes
1
answer
176
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What is the measures monad for FDHilb?
I am labouring under a particular assumption that, perhaps, needs to be corrected. I believe that FDHilb, the category of Finite Dimensional Hilbert spaces and general linear maps is a category of ...
0
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0
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In a Group as a category C with one object, How is the bifunctor ⊗ : C × C → C defined on morphisms?
I know that for the one object {1,2,3}⊗{1,2,3} = {1,2,3}. But what is (1,2) ⊗ (2,3)?
hom(A⊗B,C) ≅ hom(A,hom(B,C)) so for A = B = C
hom(C⊗C,C) ≅ hom(C,hom(C,C)), so if the group is S3 = hom(C,C), ...
6
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0
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463
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"Fundamental theorem for Hopf modules"
I am studying Hopf algebras in categories, and I hope, somebody could help me with the following.
Joost Vercruysse in his paper Hopf algebras---Variant notions and reconstruction theorems writes (...
3
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0
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153
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Is the inclusion $\Delta^{op} \to \Gamma^{op} = Fin_\ast$ homotopy cofinal?
There's a canonical functor $i: \Delta^{op} \to Fin_\ast$. For example, one uses the pullback $i^\ast$ to turn a $\Gamma$-space into a simplicial space, and then takes a geometric realization to ...
2
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0
answers
88
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Is there a notion of "graph of bundles" analogous to a graph of groups?
Since the notion of a graph of groups relies mostly on the pushout, can we construct graphs of objects in some other category, say, vector bundles? If this is the case and we have a "fundamental ...
4
votes
1
answer
449
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Krein's theorem in the Tannaka-Krein duality
In the Tannaka-Krein duality theory, the Krein theorem describes the conditions under which a given category $\varPi$ is a category of finite-dimensional representations of some compact group $G$:
...
10
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3
answers
1k
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Connections on principal bundles via stacks?
Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to [G]$, where $[G]$ is the differentiable stack ...
3
votes
0
answers
206
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Spherical perverse sheaves on the affine Grassmannian and critically twisted $D$-modules
Let $G$ be a reductive algebraic group and let $Gr_G=G((z))/G[[z]]$ be its affine Grassmannian. Define $\mathcal{D}(Gr_G)_{crit}-mod$ to be the category of right $D$-modules on $Gr_G$ twisted by the ...
3
votes
1
answer
245
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Descent in the injective model structure and descent for simplicial presheaves
In Jardine's book Local Homotopy Theory P.102, a simplicial presheaf $X$ on a site $C$ is said to satisfy descent if some local injective fibrant replacement $ j : X → Z $ is a sectionwise weak
...
0
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1
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159
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Category of Frechet Spaces is Topological?
Let $sFre_{\mathbb{R}}$ (resp. $Fre_{\mathbb{R}}$) denote the category of (resp. separable) Fr\'{e}chet spaces over $\mathbb{R}$ as objects, and bounded linear operators as morphisms.
Is this a ...
4
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0
answers
76
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Categorical construction of comodule category of FRT algebra
Let $\mathcal{B}$ denote the braid groupoid, with objects being non-negative integers $n \in \mathbb{Z}_{\geq 0}$ and morphisms $\mathcal{B}(n,n)=B_{n}$ given by the braid group. Let $\mathcal{C}$ be ...
13
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0
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433
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Examples of non-proper model structure
I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all ...
7
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329
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Hemi-semi direct product of racks or quandles
In the category of racks (similarly quandles), instead of well-known semidirect product, we have the hemi-semi direct product construction as seen on Wagemann & Crans.
As far as I know, semi ...
3
votes
1
answer
427
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Duality of Topological Vector Spaces
Let $K$ be a topological field. Let $\text{top-} K \text{-vect}$ be the category of topological $K$-vector spaces $V$, so that the maps $\cdot : K \times V \rightarrow V$ and $+ : V \times V \...
5
votes
1
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306
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Intuition behind Mac Lane's "subdivision category"
I've always felt that in proving that co/ends are co/limits, Mac Lane's CWM makes use of a category apparently coming out of nowhere.
Let $C$ be a category; I define the subdivision graph of $C$ to ...
16
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5
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Abstract nonsense versions of "combinatorial" group theory questions
In particular, I'm just curious whether there's a version of the Sylow theorems (which are very combinatorially-flavored) which allows horizontal and/or vertical categorification? Or at least can be ...
1
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0
answers
93
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Is there a well-posed definition of game on a graph? Or a well defined category of games on graphs?
All I ever found about this were natural language rules à la Asimov's three laws of robotics. The questions are straightforward questions:
1) Is there a well-posed mathematical definition of game on ...
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0
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297
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Constructions that can be seen as objects representing a functor
Some constructions can be seen as objects representing a functor.
For example,
Consider a topological group $G$ and a functor $\mathcal{F}:\text{Top}\rightarrow \text{Gpd}$ defined as $M\mapsto \...
3
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0
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84
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stack (in groupoids) over a site $\mathcal{C}$
Question : What is a stack (in groupoids) over a site $\mathcal{C}$ for you?
There are two a ways to think about it.
A stack over a site $\mathcal{C}$ is a category $\mathcal{D}$ with a functor $\...
18
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1
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716
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Is the category Idem filtered?
I have recently been reading Lurie's "Higher Topos Theory", and come upon what I believe to be an erronous claim. However, the author goes to some pain as a result of that claim, and the error seems ...
5
votes
1
answer
486
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Failure of SVC in Grothendieck toposes
The axiom SVC (for "small violations of choice") asserts that there is a set $S$ such that for every set $X$ there is a choice set $A$ such that $X$ is a subquotient of (i.e. admits a surjection from ...
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0
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Application of the cube lemma
In the paper Spherical DG-functors, the authors introduce the notion of twisted cubes and prove a lemma that they call "The cube lemma". One of the applications they prove is the following, given a ...
1
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1
answer
132
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Trying to construct the ultrafilter 2-monad on $\mathbf{Cat}$
By which I mean, following Bôrger's paper Coproducts and Ultrafilters, the terminal monad among those that preserve finite coproducts, if such a thing can be constructed.
So far, what I have is, ...
12
votes
1
answer
930
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Model existence theorem in topos theory
One of most classical and somehow striking result in classical model theory states:
A consistent first order theory $T$ has a model.
Few considerations are needed.
This result is not true for ...
2
votes
0
answers
106
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Monoidal category of irreducible highest weight modules of the Virasoro algebra
I'm trying to see if I can construct a monoidal category $\mathbf{C}$ whose objects are the irreducible unitary highest weight representations of the Virasoro algebra.
I am thinking on doing the ...
7
votes
1
answer
392
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A list of proofs of "Coherent topoi have enough points"
For my research I would like to read all the known proofs of the very classical result "Coherent topoi have enough points", by Deligne.
Ref 1: D3.3.13 in Sketches of an Elephant
provides ...
1
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1
answer
137
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Elementary example of right localization of functor
I am learning about a general framework for derived functors from Hotta et al., D-modules, Perverse Sheaves, and Representation Theory, Appendix B.
$\newcommand{\CC}{\mathcal C} \newcommand{\DD}{\...
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136
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What is the Eilenberg-Moore category for the cyclic list?
In this paper (Kock 2012), we see a data structure with circular symmetry. It is the cyclic list monad of Examples 3.10. The author is showing that data structures with symmetries can be cast as ...
1
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0
answers
172
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Injective envelope in the category of left exact functors
Let $\mathcal{A}$ be an Abelian category. $\mathcal{L}$ is the category of
absolutely pure objects of $\mathcal{A}$ and $\mathcal{L}(\mathcal{A})$ is the category of the exact left functors of $\...
10
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0
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213
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What are the (co)algebras for the $(\operatorname{Hom}(A,-), A\otimes-)$ adjunction (co)monad?
A module $A$ over a commutative ring $k$ gives a pair of adjoint endofunctors, $(A\otimes_k-)$ left adjoint to $\operatorname{Hom}_k(A,-)$. They produce a monad $T_A$ and a comonad $C_A$. Is there any ...
10
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1
answer
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Who invented Monoid?
I was trying to find (and failed) the original author of either
the concept of Monoid (set with binary associative operation and identity)
the name (which sounds french ? and also Dioid (for what ...