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Questions tagged [functors]

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2
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2answers
55 views

Is a natural transformation uniquely determined by a single morphism?

Let $C$ and $D$ be categories, let $F$ and $G$ be functors from $C$ to $D$, and let $\gamma$ and $\delta$ be natural transformations from $F$ to $G$. Then my question is, if $\gamma_a=\delta_a$ for ...
4
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2answers
39 views

Is the identity functor naturally isomorphic to a covariant dual functor?

It is often said that vector spaces are not naturally isomorphic to dual spaces, because the dual functor is not naturally isomorphic to the identity functor. But the latter is a rather trivial ...
0
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1answer
34 views

Can natural transformations be viewed as functors between images of functors?

Let $C$ and $D$ be categories, and let $F,G:C\rightarrow D$ be functors. Then a natural transformation $\tau$ from $F$ to $G$ is a family of morphisms $\{\tau_x\}_{x\in C}$ where for each $x\in X$, $\...
0
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1answer
36 views

Equivalent definitions of a contravariant functor

Let $\mathcal C,\mathcal D$ be categories. A contravariant functor from $\mathcal C$ to $\mathcal D$ is a functor $F$ from $\mathcal C^{op}$ to $\mathcal D$ . Another definition is given on Wikipedia: ...
3
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0answers
86 views

Homology of the complex of exact sequences of homotopy groups

By naturality of the connecting homomorphism, relative homotopy $\pi_{\bullet}$ may be regarded as a functor from the category of pointed pairs of topological spaces to the category of exact sequences ...
4
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0answers
33 views

Proof verification: An arrow which is monic under a faithful functor is itself monic

Context: To introduce some symbols and such, what I'm seeking to prove is this: Let $F$ be a faithful functor. Suppose $F(f)$ is a monic arrow. Show $f$ is monic. This came up as part of a class ...
2
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2answers
63 views

Can I define a functor F and a “ΔF” of sorts, which will uniquely determine a new functor?

Let $F: \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Natural transformation between $F$ and some other functor is defined as an assignment of a morphism in $\mathcal{D}$ to each object in $\...
2
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0answers
48 views

Well-powered/subobject functor?

Is there a construction which maps each object in a category to the set of its subobjects? Concretely, I'm interested in mapping an object $M$ in the category of manifolds $\mathbf{Man^1}$ to the set ...
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0answers
25 views

Adjoint functors for Lie Algebras

Let's restrict to finite dimensional case. Functor $(-)^{\mathrm{ab}}: \mathrm{LieAlg} \to \mathrm{AbLieAlg}$ is left adjoint of the inclusion functor $i: \mathrm{AbLieAlg} \to \mathrm{LieAlg}$. ...
6
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1answer
101 views

Showing that $F$ is not representable [closed]

As I'm trying to find (counter)examples of representable functors, I tried looking up some instructive examples. One of the counterexamples I'm having trouble with, is the following: Show that the ...
7
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5answers
135 views

What constructions of “elementary” mathematics are actually functors?

I'm not looking for the usual simple examples of functors like the fundamental group or forgetful functors, what I'm looking for is some interesting examples of constructions from "elementary" ...
2
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1answer
22 views

Is additivity necessary for a left exact functor to preserve pullbacks?

I'm having a bit of difficulty with exercise 5.16 from Rotman's An Introduction to Homological Algebra (second edition). The exercise (at least the relevant part) reads Prove that every left exact ...
2
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1answer
48 views

How is it obvious that $\times : C \times C \to C$ is right adjoint to the diagonal functor?

This is from "Sheaves in Geometry & Logic". $\times : C \times C \to C$ is the cartesian product of two objects. So assume that finite products exist in $C$ the above is a functor. To say ...
2
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1answer
44 views

checking the functor $\texttt{Nil}_n$ is represented by $(\mathbb{Z}[x]/(x^n), \tau_R)$

This is the continuation of another question I did some days ago. Here. I have been working on it and I would like to know if my try to prove it is right or not. I would appreciate a lot any feedback ...
3
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1answer
28 views

What does the endofunctor/monad that sends a set to the set of finite words on the set do to morphisms?

Suppose we have a monad $T:Set \rightarrow Set$ that sends a set X to the set of finite words on the set X, with the unit and multiplication being inclusion and concatenation respectively. What does ...
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0answers
31 views

Join of simplicial sets induces a functor.

Given a simplicial set $X$, denote by $(\mathrm{Set}_{\Delta})/_X$ the over category, whose objects are morphisms of simplicial sets with source $X$. I want to show that the joint $X \star Y$ of ...
1
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1answer
67 views

checking that there is a functor $\texttt{Nil}_n: Ring \longrightarrow Set$

I want to show that $\forall n \geq 1$, there is a covariant functor $\texttt{Nil}_n: Ring \longrightarrow Set$ that sends a ring $R$ to the set $\{x\in R | x^n = 0\}$. I have being thinking that the ...
1
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2answers
99 views

$\mathcal{Nil}_n$ isomorphic to $h^A$ with $A=\mathbb{Z}[x]/(x^n)$

I would like to see that $\texttt{Nil}_n$ is isomorphic to $h^{\mathbb{Z}[x]/(x^n)}$ as categories. $\texttt{Nil}_n: \texttt{Rings} \longrightarrow \texttt{Sets}$ is the functor that sends a ring $R$ ...
1
vote
1answer
71 views

Why is $\mathbb{G}_m$ is a representable functor?

What does it mean that multiplication $\mathbb{G}_m$ is a representable functor, with $\mathbb{G}_m = \text{Spec}(\mathbb{Z}[x,x^{-1}])$ ? When I looked at the stacks project page on $\mathbb{G}_m$ ...
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0answers
38 views

Relation between functors under an equivalence

Let $A$, $B$, and $C$ be three categories and $G\colon A\to C$ and $H\colon B\to C$ two functors. Assume that $F\colon A\to B$ and $E\colon B\to A$ form an equivalence between $A$ and $B$. Suppose ...
2
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2answers
64 views

Product in the category of functors.

Let $A$ be a category and $C= Fun(A, Set)$ (i.e. the objects are functors and morphisms are natural transformations between them). I want to know if this category has a product. For given $X \in A$ ...
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0answers
38 views

$[\mathsf{I},\mathsf{C}]$ and $[\mathsf{J},\mathsf{C}]$ are equivalent if $\mathsf{I}$ and $\mathsf{J}$ are. [duplicate]

Let $\mathsf{I}$ and $\mathsf{J}$ be equivalent categories. Let $\mathsf{C}$ be another category. I need to prove that the categories of functors $[\mathsf{I},\mathsf{C}]$ and $[\mathsf{J},\mathsf{C}]$...
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0answers
74 views

Model category of all model categories

Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What ...
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0answers
74 views

What is an algebra over a monad?

The category of algebras over a monad (also: “modules over a monad”) is traditionally called its Eilenberg–Moore category (EM) In that context What exactly does the word "Algebra" mean? What ...
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0answers
40 views

Sandwich natural transformation between two functors

In the Kleisli adjunction we have: $G\varepsilon F = \mu$ where $\varepsilon$ is a natural transformation called the counit. How exactly is $G\varepsilon F$ defined? I understand $G\varepsilon$ and $...
2
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1answer
60 views

If the Yoneda lemma essentially says that $\text{Hom}(\text{Hom}(\cdot, x), A) \simeq A(x)$, then what about higher iterates of $\text{Hom}$?

Assume that the $C$ in $\text{Hom}_C(x,y)$ can always be inferred from $x,y$ so that we can change our notation to $\text{H}(x,y) := \text{Hom}_C(x,y)$ Then the Yoneda lemma "looks at a single step ...
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0answers
29 views

How to arrive at unique factorization through the limit given naturality compatibility conditions?

If $\alpha : I \to C$ from a small category to any category $C$. Define a functor $\lim\limits_{\rightarrow} \alpha : X \mapsto \lim\limits_{\leftarrow} \text{Hom}_C(\alpha, X)$ from $C^{op}$ to $\...
1
vote
1answer
41 views

Definition of symmetric power of a linear represenation

I'm reading Kowalski's Representation theory, and there's a part about the symmetric and antisymmetric powers of a representation, and I'd like to ask a question about those. So there's a proposition ...
1
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1answer
30 views

Showing when two functors are naturally isomorphic, if one is faithful, then the other also is. [duplicate]

Supposing we have a natural isomorphism $\tau : S \rightarrow T$ between functors $S,T : \mathscr{C} \rightarrow \mathscr{D}$, how exactly do we show that if $S$ is faithful, then so is $T$? If $S$ ...
0
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2answers
43 views

Proving for two naturally isomorphic functors, if one is full, then so is the other. [duplicate]

So we let $S,T : \mathscr{C} \rightarrow \mathscr{D}$ be naturally isomorphic functors. We seek to show that if $S$ is a full functor, then so is $T$. As given, we have a natural isomorphism $\tau : ...
0
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1answer
31 views

Right whiskering, unknown naturality

Suppose $F,G:C\to D$ and $H:D\to E$ are functors and $\alpha:F\to G$ is a natural transformation.Let $H\circ \alpha:H\circ F\to H\circ G$ be the right whiskering $(H\circ \alpha)_A:H(FA)\to H(GA)$ ...
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0answers
81 views

Nearly locally presentable categories

Here1, in the remark $2.3 (1)$ how from the fact that ${\cal K}(A,-)$ does not preserve coproducts it follows that ${\cal K}(A,-)$ sends special $\lambda$-directed colimits to $\lambda$-directed ...
2
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1answer
55 views

A bijection $\mathsf{Ob}([\mathsf{C\times D},\mathsf{E}])\to\mathsf{Ob}([\mathsf{C},[\mathsf{D},\mathsf{E}]])$

Let $\mathsf{C},\mathsf{D}$ and $\mathsf{E}$ be categories. Consider a function $\rho\colon\mathsf{Ob}([\mathsf{C\times D},\mathsf{E}])\to\mathsf{Ob}([\mathsf{C},[\mathsf{D},\mathsf{E}]])$ which maps ...
2
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1answer
36 views

An exercise about bifunctors from Riehl's “Category Theory in Context”

This is an exercise from E.Riehl's book "Category Theory in Context" (p.48, ex.1.7.vii) Prove that a bifunctors $F\colon\mathsf{C}\times\mathsf{D}\to\mathsf{E}$ determines and is uniquely ...
0
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1answer
31 views

How do you arrive at $\eta : \text{Hom}_{C'}(Y, Y') \to \text{Hom}_{C'}(LR(Y), Y')$ from $\eta : LR \to \text{id}_{C'}$?

On page 28 of "Categories and Sheaves" it says: $$ \eta : L R \to \text{id}_{C'} $$ is a functor but then they have in a commutative diagram right below that: $$ \text{Hom}_{C'}(Y, Y') \xrightarrow{...
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0answers
46 views

If $F$ and $G$ are adjoint functors, why are $\operatorname{Nat}(G,G)\simeq\operatorname{Nat}(F,F)$ as algebras? [duplicate]

Suppose $F\colon\mathcal{A}\to\mathcal{B}$ and $G\colon\mathcal{B}\to\mathcal{A}$ are adjoint functors between some $R$-linear abelian categories ($R$ a ring), via a fixed counit-unit adjunction $\...
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0answers
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In the proof of the Yoneda Lemma in “Categories & Sheaves” by Kashiwara & Schapira.

Our goal is to show that $\text{Hom}_{C^{\wedge}}(h_C(X), A) \simeq A(X)$. We first want to show a map from left to right. The book says: $$ \text{Hom}_{C^{\wedge}}(h_C(X), A) \to \text{Hom}_{\text{...
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1answer
96 views

Does the functor $V\mapsto V^{**}/V$ (for $\dim V=\infty$) reflect monomorphisms, epimorphisms or isomorphisms?

Let $\textbf{Vect}_\infty$ be the category of infinite dimensional vector spaces (over some fixed ground field), and consider the endofunctor $F$ of $\textbf{Vect}_\infty$ whose effect on objects is ...
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0answers
26 views

How to prove naturality in $c$ of $\text{Hom}_D(Fc, d) \simeq \text{Hom}_C(c, Ud)$ given adjunction $(F, U)$?

We're given an adjunction $(F, U)$ where $F: C \to D, U : D \to C$ are the left and right adjoint functors. We use the definition that the adjunction comes with a "unit of adjunction", or a natural ...
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0answers
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The category of functors if the codomain category has zero object is non-empty. Why?

Here, page 6, Daniel Murfet said, the category of functors $(\mathcal{A}, \mathcal{B})$ can be empty, although it is nonempty if $\mathcal{B}$ has zero object. Why? [Zero object: An object which is ...
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0answers
37 views

Naturality of Conditional Expectation

Context Let $\mathscr{G}$ be a $\sigma$-subalgebra of $\mathscr{F}$ and let $\mathbb{P}$ be a probability measure on a polish-space $X$. Define the category $\mathfrak{C}$ of random-elements in $L^...
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0answers
62 views

What is the definition of a localization of a category?

There appears to be a discrepancy in the literature regarding the definition of a localisation of a category. Let $\mathcal{C}$ be a category and let $S$ be a class of morphisms. The classical ...
3
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1answer
64 views

Does “inverting the corresponding morphisms” in a functor have a name?

Let $F:C\to D$ be a functor between categories $C$ and $D$. Assume that for every morphism $f:X\to Y$ in $C$, the corresponding morphism $F(f):F(X)\to F(Y)$ is invertible. Let $G(f)$ be the inverse of ...
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1answer
65 views

The category of Z-modules is isomorphic to the category Ab

How to show that the category of $\mathbb{Z}$-modules is isomorphic to the catefory of abelian groups. It seems obvious for me that a $\mathbb{Z}$-module is an abelian group and conversely. But I don'...
2
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2answers
71 views

Infinite limit of finitary functors

I'm learning about finitary functors. I have two conflicting results: $F_A X = X^A$ is finitary iff $A$ is finite. Therefore, $X^\mathbb{N}$ is not finitary. $F X = X$ is finitary and the category of ...
2
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0answers
44 views

Right/Left-exactness in Abelian categories [duplicate]

All definitions I found say that a functor $F:\mathcal{A}\rightarrow\mathcal{B}$ is right-exact, if for every short exact sequence $$ 0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$$ gives ...
7
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1answer
92 views

Mistake in Mac Lane's presentation of the “universal natural transformation”?

On page 39 of Category Theory for Working Mathematicians, Mac Lane makes a claim that seems to me to be false. Let $\mathcal{C}$ be a category, and let $\mathbf{2}$ be the category with exactly two ...
2
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1answer
38 views

Precondition “small category” in functor category

I am currently working on a exercise sheet about categories. There are two exercises: In the first parts I have to show that the vertical composition and the horizontal composition of two natural ...
0
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1answer
42 views

On actions and functors

Let $A$ be an algebra and $f:A\to A$ an automorphism of $A$. Further, we denote the category of finite-dimensional left $A$-modules with $A$-mod. $f$ defines a functor $F:A\text{-mod}\to A\text{-mod}$...
4
votes
1answer
159 views

Commuting diagram for (monoidal) functors

Assume we have two categories $\cal A,B$ and four functors $F,G:\cal A\to B$, $D:\cal A\to A$, $E:\cal B\to B$. In order to show an equality like $$E\circ F=G\circ D\quad (1)$$ I have to prove that ...