Questions tagged [functoriality]
The functoriality tag has no usage guidance.
31
questions
14
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5
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Birkhoff's representation theorem vs matroid-geometric lattice correspondence
This question is motivated by the superficial observation that Birkhoff's representation theorem and the cryptomorphism between matroids and geometric lattices are sort of similar. The former says ...
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5
votes
1
answer
130
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Exactness of functors in a $C^*$-tensor category
I'm reading the book "Compact quantum groups and their representation categories" by Neshveyev-Tuset. In section 2.3 "Fiber functors and reconstruction theorems*, the following ...
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0
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How to check the non-emptyness of the A-packet of non-split $\operatorname{SO}(2n+1)$?
Let $F$ be a number field and $G_n=\operatorname{SO}(2n+1)$ be the split group over $F$. Let $G_n^{\times}$ be a non-split group over $F$.
Let $\tau$ be an irreducible cuspidal automorphic ...
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votes
2
answers
554
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Is the injective envelope functorial?
Let $A$ and $B$ be unital $C^*$-algebras, so we can view these as operator systems, and it makes sense to consider their injective envelopes $I(A)$ and $I(B)$. These injective envelopes become $C^*$-...
- 83
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2
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Subobjects as an object in a topos
Forgive me if this question turns out to be too elementary-then feel free to move it to stack exchange. I believe that this should be very basic fact from topos theory nevertheless being not familiar ...
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4
votes
1
answer
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Linearity of covariant and contravariant $Ext^1$ functors defined via short exact sequences
Let $R$ be a Commutative ring. Let $M,X,Y$ be $R$-modules. Let $f: X \to Y$ be an $R$-linear map.
Then, given an exact sequence $\eta: 0\to X \to Z_{\eta} \to M \to 0$ in $Ext^1(M,X)$, the pushout of $...
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votes
1
answer
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Classification of the functors on the category of cyclic groups
Let $\mathsf{Grp}$ be the category of groups and let $\mathsf{Cyc}$ be the subcategory of cyclic groups.
As seen in the posts here and there (and their answers), a functor $F: \mathsf{Cyc} \to \mathsf{...
- 24.7k
7
votes
0
answers
166
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Open subfunctor of Quot Functor induced by open immersion
Let $f: X \rightarrow S$ be a morphism of noetherian schemes and $\mathfrak{Q}uot_{\mathcal{E}/X/S}$ be the functor parametrizing families of quotients of $\mathcal{E}$ in the category of locally ...
- 171
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2
answers
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What structure do natural isomorphisms preserve?
My understanding from model theory is that, given groups A and B, the statement $ A \cong B $ implies that any for any first order statement $P$ in the language of groups, $P(A) \iff P(B) $.
Can an ...
- 111
10
votes
1
answer
508
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Functoriality of the Hopf dual
Given Hopf $\mathbb{C}$-algebra $H$, it's Hopf dual $H^o$ is the largest Hopf algebra contained in $H^*$, the $\mathbb{C}$-linear dual of $H$. (This is well known to be well-defined, see for example ...
- 805
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Does each monotonic endofunctor on the category of sets and relations preserve conversion?
Consider a functor $F : \mathbf{Rel} \to \mathbf{Rel}$ that is monotonic (for all relations $R$ and $S$ with $R \subseteq S$ we have $FR \subseteq FS$). Does such a functor always preserve conversion ...
- 951
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156
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Functoriality and proofs
Recently, I began to study $\mathcal{D}$-modules and one of the major problems I encountered is that many natural transformations on the (derived) category of $\mathcal{D}$-modules are defined as the ...
- 1,295
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320
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Coinflation in cohomology
Let $U$ be a normal subgroup of a group $G$ of finite index. On cohomology, somewhat dual to the functorially defined restriction map, $\text{res}^G_U\colon H^n(G, A) \to H^n(U, A)$, the finite index ...
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988
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Where stands functoriality in 2017?
In 2002, R. Langlands put forward a new strategy to prove the general functoriality conjecture in the Beyond endoscopy paper. The main purpose of this strategy is to detecting the automorphic ...
11
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Explicit L-factor for supercuspidals
I am interested in L-functions for the quasi-split unitary group $U$ in three variables, following the construction by zeta integrals of Gelbart-Piatetski-Shapiro. My aim is to understand the ...
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1
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471
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Loss of cuspidality by Langlands tranfer
Given an $L$-homomorphism of Langlands dual groups
$${}^LG \to {}^LG'$$
Langlands functoriality contectures predicts the existence of a tranfer map of automorphic representations
$$Aut(G) \to Aut(G')$...
- 541
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1
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400
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Functorial description of mod-2 homology of an abelian group $A$ in terms of $A/2$ and ${}_2A.$
Let $A$ be an abelian group and $p$ be a prime. If $p\ne 2,$ there is a very nice functorial description of the homology algebra $H_*(A,\mathbb Z/p):$
$$H_*(A,\mathbb Z/p)\cong \Lambda^*(A/p)\otimes \...
- 1,282
4
votes
1
answer
255
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Equivalence of two pictures of odd $K$-theory
One can show that two functors $K^0$ and $K_0(C(-))$ from the category of compact topological spaces to the category of abelian groups are naturally equivalent. The first one is topological $K$-theory ...
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8
votes
1
answer
312
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What's the (monoidal) image of a monoidal functor?
For an ordinary functor $F\colon \mathcal{C} \to \mathcal{D}$ of categories, there is a construction $\operatorname{im} F$, the image of $F$, which is again a category, and $F$ factors through that ...
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votes
3
answers
188
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Set of functions is not a bifunctor on Rel
Let Rel be the category whose objects are sets and whose morphisms are binary relations, with composition defined by $x (S \circ R) z \Leftrightarrow (\exists y : x R y \wedge y S z)$, and identity ...
- 429
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votes
1
answer
235
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Action of a strict 2-group on a category gives autoequivalences?
A strict action of a strict 2-group (seen as 2-category with only one object $\star$) $\mathcal{G}$ on a 2-space (principally a category) $\mathcal{M}$ is a strict 2-functor $\Phi:\mathcal{G}\to \...
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Which properties determine the uniqueness of the local Artin map?
Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...
- 5,898
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votes
1
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326
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Functoriality of the adjoint functor construction?
Say we have a category $\mathcal C$, and for every category $\mathcal A$, we have a category $\mathcal D_{\mathcal A}$ and a functor $F_{\mathcal A} : \mathcal D_{\mathcal A} \to \mathcal C$, and, ...
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votes
2
answers
7k
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Current Status on Langlands Program
The Langlands Program was launched almost fifty years ago, and progress has been made gradually, much of it hard earned. Langlands himself wrote a survey on the functoriality conjecture in 1997, Where ...
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15
votes
2
answers
2k
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The status of automorphic induction
Background: Let $K/F$ be a degree $r$ extension of number fields. It is conjectured that an automorphic representation of GL$_n$ associated to $K$ induces an automorphic representation of GL$_{rn}$ ...
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1
answer
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Functoriality of the cotangent bundle
Recall that to any manifold $X$, I can assign in a canonical way a manifold $\mathrm T ^* X$, the total space of the cotangent bundle over $X$. Recall also that, unlike the tangent bundle ...
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1
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527
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Functoriality of a standard integral domain construction.
The evident forgetful functor from fields to integral domains has a left adjoint, namely the construction of the quotient field for a given integral domain. Another standard construction is taking the ...
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answers
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Functorial point of view of spectrum (Looking for reference)
I think I should elaborate a bit. What I am asking is the definition of spectrum of a category as a stack in functor view of points.
In noncommutative algebraic geometry. We define spectrum of an ...
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366
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Is there functorial point of view to differential operator?
This question is related to differential operator in noncommutative geometry. I wonder whether there is any approach to differential operator that taking differential operator as a functor? I think it ...
- 1,235
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votes
2
answers
786
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How to distinguish between natural and unnatural equivalences of categories
Some equivalences of categories are constructed by explicitly giving a pair of functors that are inverses up to isomorphism. For example, the equivalence between CRing^op and affine schemes is given ...
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5
answers
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Where stands functoriality in 2009?
Robert Langlands is famous in number theory for making famous and deep conjectures about very abstract things called automorphic forms, somewhere in the 60s.
There's a very interesting article by ...
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