# Questions tagged [functoriality]

The functoriality tag has no usage guidance.

40
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### A distributor between categories induces a distributor between their categories of presheaves

Let $P$ be a distributor/profunctor from a small category $A$ to a small category $B$, i.e. a functor $P : B^\circ \times A \to \mathrm{Set}$.
We may then define a distributor from $[A^\circ, \mathrm{...

4
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1
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103
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### Are (commutative) squares in some sense universal among edge-symmetric double categories?

Definitions:
Given a category $\mathcal{C}$, and a double category $\mathbb{D}$, i.e. a category internal to $Cat$ with an “objects category” $\mathcal{D}_0$ and a “morphisms category” $\mathcal{D}_1$,...

6
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1
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### Is univalence equivalent to every type function being a functor over equivalence?

Introduce a rule in type theory that if $\Gamma \vdash f : \text{Type} \to \text{Type}$ and $\Gamma \vdash e : A \simeq B$ then $\Gamma \vdash f[e] : f(A) \simeq f(B)$.
It may seem like such a rule is ...

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### Does the symmetric algebra functor preserve inclusions?

Theorem: For any compact abelian group $G$, the homogeneous component $%
H^{2}\left( B_{G};%
%TCIMACRO{\U{2124} }%
%BeginExpansion
\mathbb{Z}
%EndExpansion
\right) $ of degree $2$ is naturally ...

5
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1
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### 3-functoriality of the lax Gray tensor product

In Formal category theory: adjointness for 2-categories, Gray defines a tensor product of 2-categories, now more commonly known as the lax Gray tensor product, which I will denote by $\otimes_l$. For ...

-1
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1
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167
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### Does a symmetric monoidal functor between cartesian monoidal categories automatically preserve products? [closed]

Suppose $\cal A, B$ are cartesian monoidal categories, that is, categories equipped with a choice of finite products (including the nullary one, $1$). Suppose, moreover, that $F: \cal A \to B$ is a ...

11
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2
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### Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property?

I am wondering if the category of small categories $\mathbf{Cat}$ is known to (not) have the Cantor–Schröder–Bernstein property? That is, for any two categories $\mathcal{C}$ and $\mathcal{D}$, does ...

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### Is Landvogt's thesis "The functorial properties of the Bruhat–Tits building" available online?

Universität Münster publishes theses online through "miami", but "miami" doesn't have Erasmus Landvogt's thesis (search).
ProQuest (predatorily) provides many theses, but they don'...

1
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1
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92
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### Find a functorial zig-zag of spaces

This is a rather broad question. Suppose you have an ordinary category $C$ (for example, $\Delta$), and two diagrams $X_{\bullet}, Y_{\bullet} : C \to \textrm{Top}$. Suppose also that $X_c$ is ...

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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 ...

5
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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 ...

6
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2
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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 $C^*$-...

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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 ...

4
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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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1
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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{...

7
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0
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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 ...

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2
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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 ...

10
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1
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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 ...

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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 ...

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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 ...

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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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### 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 ...

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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 ...

11
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1
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483
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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')$...

11
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1
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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 \...

5
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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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### 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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### 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
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1
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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 ...

3
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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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### 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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### 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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### 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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### 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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### 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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### 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
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2
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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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### 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 ...