Questions tagged [functors]
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2
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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
vote
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
vote
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 $\mathbb{G}_m$ ...
0
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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
votes
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 $X \in A$ ...
1
vote
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 $[\mathsf{J},\mathsf{C}]$...
4
votes
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 ...
0
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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 ...
0
votes
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
votes
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 ...
1
vote
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
vote
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
votes
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
votes
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 $(H\circ \alpha)_A:H(FA)\to H(GA)$ ...
1
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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
votes
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
votes
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
votes
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:
$$
\text{Hom}_{C'}(Y, Y') \xrightarrow{...
3
votes
0answers
46 views
If $F$ and $G$ are adjoint functors, why are $\operatorname{Nat}(G,G)\simeq\operatorname{Nat}(F,F)$ as algebras? [duplicate]
Suppose $F\colon\mathcal{A}\to\mathcal{B}$ and $G\colon\mathcal{B}\to\mathcal{A}$ are adjoint functors between some $R$-linear abelian categories ($R$ a ring), via a fixed counit-unit adjunction $\...
0
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0answers
43 views
In the proof of the Yoneda Lemma in “Categories & Sheaves” by Kashiwara & Schapira.
Our goal is to show that $\text{Hom}_{C^{\wedge}}(h_C(X), A) \simeq A(X)$.
We first want to show a map from left to right.
The book says:
$$
\text{Hom}_{C^{\wedge}}(h_C(X), A) \to \text{Hom}_{\text{...
5
votes
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 ...
0
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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 ...
1
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0answers
29 views
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 ...
2
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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^...
3
votes
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
votes
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 ...
1
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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 ...
