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Questions tagged [functoriality]

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10
votes
1answer
412 views

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 ...
7
votes
0answers
142 views

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 ...
5
votes
0answers
110 views

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 ...
3
votes
0answers
131 views

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 ...
10
votes
0answers
682 views

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 ...
10
votes
0answers
144 views

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 ...
11
votes
1answer
409 views

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')$...
11
votes
1answer
359 views

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 \...
4
votes
1answer
234 views

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 ...
6
votes
1answer
216 views

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 ...
5
votes
3answers
154 views

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 ...
3
votes
1answer
173 views

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 \...
0
votes
0answers
64 views

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 ...
3
votes
1answer
239 views

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, ...
34
votes
2answers
5k views

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 ...
15
votes
2answers
963 views

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$_{rn}$ ...
17
votes
1answer
958 views

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 ...
1
vote
0answers
426 views

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 ...
7
votes
0answers
469 views

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 ...
1
vote
0answers
318 views

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 ...
9
votes
2answers
738 views

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 ...
26
votes
5answers
4k views

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 ...