# Questions tagged [functoriality]

The functoriality tag has no usage guidance.

25
questions

**8**

votes

**1**answer

343 views

### 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{...

**6**

votes

**0**answers

105 views

### 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 ...

**10**

votes

**2**answers

713 views

### 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 ...

**10**

votes

**1**answer

474 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

**0**answers

143 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

**0**answers

134 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

**0**answers

218 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

**0**answers

809 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

**0**answers

166 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

**1**answer

439 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

**1**answer

375 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

**1**answer

240 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 ...

**8**

votes

**1**answer

246 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

**3**answers

175 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

**1**answer

190 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

**0**answers

70 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

**1**answer

273 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, ...

**35**

votes

**2**answers

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

**2**answers

1k 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$_{rn}$ ...

**18**

votes

**1**answer

1k 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

**0**answers

468 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

**0**answers

504 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

**0**answers

339 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

**2**answers

753 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

**5**answers

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 ...