Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

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Understanding the determinant example of a natural transformation.

This question is about intuition. In the example below, how do we understand the natural transformation $\det$ as a relationship between the two functors given rather than as a relationship between ...
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List of Naturally Isomorphic Real Vector Space Pairs

Let's consider the category of finite dimensional real vector spaces (VS) / inner product spaces (IPS). Which of the following pairs of isomorphic vector spaces (given appropriate dim constraints), ...
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Unitors in star-autonomous categories

1.Context Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume that there are bijections $\phi_{X,Y,...
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Do the components of a natural transformation need to be part of the category?

Given two functors, $F$ and $G$, between categories $\mathbf{C}$ and $\mathbf{D}$, a natural transformation $\eta$ associates a morphism $\eta_X$ for every $X$ in $\mathbf{C}$. This morphism is ...
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Equality of morphisms in star-autonomous categories

1.Context Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume that there are bijections $\phi_{X,Y,...
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How to find more (maybe: all) natural isomorphisms with vector space and tensor constructions

Background: I am refreshing my knowledge of tensor dualities to catch up with some physical applications. Example 1: I am aware that $\mbox{Hom}(V, \mbox{Hom}(W,U))$ is naturally isomorphic to $\mbox{...
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35 views

Weighted limits

I have a very trivial question about the page 80 here: how this shape of $W$ $$W:2\to \mathbf{Set}$$ with $$\ast\sqcup\ast\to \ast$$ implies that the components of $$W\Rightarrow\cal{M}(m,f)$$ are ...
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The only natural transformation $\text{id}_\mathsf{Ring}\to\text{id}_\mathsf{Ring}$ is the identity

I want to show that for any natural transformation $\eta:\text{id}_\mathsf{Ring}\to\text{id}_\mathsf{Ring}$ we have that $\eta_R=\text{id}_R$ for all $R\in\text{ob}(\mathsf{Ring})$. I was able to show ...
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Do (special) natural transformations imply a commutative triangle of functors (but NOT vice versa)?

Given categories $\mathcal{C}, \mathcal{D}$, let functors $F,G: \mathcal{C} \to \mathcal{D}$ be such that for any objects $C_1, C_2 \in Ob(\mathcal{C})$, $F(C_1) = F(C_2) \implies G(C_1) = G(C_2)$, ...
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Isomorphism not natural in $X$?

$\require{AMScd}$ I am working on the following task: Let $\mathcal{H}_*$ be a homology theory and let $X \neq \emptyset$ be a space. Construct an isomorphism $\mathcal{H}_n(X) \cong \widetilde{\...
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Is natural isomorphism unique?

suppose $F,G\in Ob(Fct(\mathscr C_1,\mathscr C_2))$ are functors,and $\theta:F\Leftrightarrow G$ is an natural isomorphism between $F$ and $G$ my question is : Are there any other $\theta':F\...
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In what sense is the uniqueness of left adjoint isomorphism 'canonical'

In my category theory course, Peter Johnstone has written that for any two left adjoints $F$, $F'$ "there is a canonical natural isomorphism $F \to F'$" Explicitly, this isomorphism is that ...
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3 votes
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Any two natural transformations between identity functors commute

Let $\mathcal{C}$ be a category, $id_\mathcal{C}:\mathcal{C} \to \mathcal{C}$ the identity functor. Prove that for any two natural transformations $\alpha, \beta : id_\mathcal{C} \Rightarrow id_\...
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Natural transformations of the identity functor

For a category $\mathbb{C},$ with identity functor $\mathbb{C} \overset{1_\mathbb{C}}{\rightarrow} \mathbb{C},$ I have recently found that natural transformations $1_\mathbb{C} \overset{\alpha}{\...
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Morphism between free abelian groups is 1-1; Vick Prop. 1.9

Proposition 1.6 in Vick's Homology Theory states: If $X$ is a space and $\{X_{\alpha}:\alpha \in A\}$ are the path components of $X$, then $$H_{k}(X) \approx \sum_{\alpha \in A}H_{k}(X_{\alpha}).$$ ...
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Natural Transformation of Bifunctors

I had a hard time proving the statement: "a transformation between two bifunctors is natural if and only if it is a natural transformation in each of it's arguments". This is Proposition no. ...
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$n$-Simplices of Fiber product of Simplicial sets

Let $X,Y Z$ be simplicial sets with two maps $a: X \to Z, b: Y \to Z$ and I would like discuss how the $n$-simplices of the new simplicial set $X \times_Z Y$. I found only a construction for direct ...
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1 vote
1 answer
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Why $\Psi\circ\Phi=1$ in Yoneda lemma?

I am trying to understand the proof: How can one show that $\Psi\circ\Phi=1_{\textrm{Nat}(A(A,-),F)}$? Let $\tau:A(A,-)\rightarrow F$ be a natural transformation. Then $(\Psi\circ\Phi)(\tau)=\Psi(\...
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2 answers
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Is the standard isomorphism $L(F^n,F^n)\to F^{n\times n}$ a natural transformation?

Let $V$ be an $n$-dimensional vector space over a field $F$. I was wondering if the algebra isomorphism $$\Phi\colon L(F^n,F^n)\to F^{n\times n}$$ defined by $$A(x)=\Phi(A)\cdot x\quad\forall x\in F^n$...
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Change scores/difference scores on a logarithmic scale

From a longitudinal cohort-study I have two datapoints per patient on arterial calcification ($x_{new}$ and $x_{old}$). To determine risk factors for calcification-change I want to use a linear ...
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Square of unit in a monoidal category

Suppose we have a monoidal category $(C,\otimes,I)$ with left and right unitor being $\lambda$ and $\rho$. They yield two morphisms $\lambda_I,\rho_I:I\otimes I\to I$. It seems to me that both ...
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Natural transformation on $H$-groups

I'm currently reading Spanier "algebraic topology". Let $\alpha:P\to P'$ be a map between $H$-groups. Then $\alpha_\sharp$ is a natural transformation from $\pi^P$ to $\pi^{P'}$ in the ...
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Definition of an algebra over a monad by using equalities between natural transformations

In the definition of a monad, there are two ways to specify the equations: equalities between natural transformations, or equalities between morphisms as is done there on Wikipedia. In the usual ...
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A natural isomorphism $\mathsf C(\operatorname{colim} F, Y) \cong \lim \mathsf C(F-, Y)$. Understanding $\lim$ as a functor?

In the text, "Topology: A Categorical Approach", there is an exercise generalizing the identity $\mathsf C(\coprod X_i, Y) \cong \prod \mathsf C(X_i, Y)$: Prove that a functor $F: \mathsf B ...
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1 answer
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Hilton/Stammbach Exercise 2.4.6: Yoneda Embedding and Functors

I'm working on the following exercise from Chapter 2 of Hilton/Stammbach's A Course in Homological Algebra: "Let $\mathfrak{A}$ be a small category and let $Y : \mathfrak{A} \to [\mathfrak{A}^\...
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Question about a statement in Mac Lane's "Categories for the working mathematician" about a bijection being natural in an object

I have a question about a statement in Mac Lane's "Categories for the working mathematician". It is in page 51, in the context of graphs and free categories. The statement basically says: ...
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Number of Natural Transformation

I am working on the following problem: Let $F:\text{Set}\to\text{Set}$ be the functor that has the object map $X\to X\times X$ and the morphism map $(f:X\to Y)\to (f\times f:X\times X\to Y\times Y)$ ...
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2 answers
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On a special kind of $R$-linear functor from an $R$-linear additive category [closed]

Let $R$ be a Commutative Noetherian ring, let $\mod (R)$ be the category of finitely generated $R$-modules. Let $\mathcal C \subseteq \mod(R)$ be an $R$-linear (https://stacks.math.columbia.edu/tag/...
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2 votes
1 answer
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Show that $Sh(C,J)^{D^{op}}$ is a Grothendieck topos

Let $C$ be a small category and $J$ a Grothendieck topology on $C$. Let $Sh(C,J)^{D^{op}}$ be the category of functors $D^{op}\rightarrow Sh(C,J)$ and natural transformations between them, for some ...
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2 votes
1 answer
101 views

Definition of $\tilde{H}_n(X,A)$

My question arises from the following sentence of Hatcher's book p.118, in particular I do understand that $\tilde{H}_n(X,A)$ is defined to be $H_n(X,A)$ if $n \ne 0$. There is a canonical way to ...
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4 votes
1 answer
212 views

pair homotopic maps induce the same homology

I'd like to prove that given $f,g : (X,A) \longrightarrow (Y,B)$ homotopic as map of pair, i.e $H(A\ \times I) \subset B$ then they induce the same homology. I already know the theorem which states ...
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1 vote
2 answers
60 views

Does taking restriction and left derived functors commute?

Let $R$ be a commutative Noetherian ring. Let Mod$(R)$ denote the category of $R$-modules, and mod$(R)$ denote the category of finitely generated $R$-modules; notice that both of these categories are ...
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7 votes
1 answer
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Yoneda's lemma: group morphisms give Hopf-algebra morphisms

Let $k$ be a commutative ring. Let $\text{Alg}$ be the category of commutative $k$-algebras and $\text{CHopf}$ the category of commutative Hopf-algebras. Let us also write $[\text{Alg}, \text{Grp}]$ ...
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2 votes
1 answer
152 views

Tensor products and natural isomorphisms

Let $U$ and $V$ be two finite-dimensional vector spaces. How to prove that $U \otimes V \cong V \otimes U$? The equivalence symbol generally refers to natural isomorphisms – i.e. isomorphisms defined ...
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1 answer
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Definition of functor in equivalence with skeleton

In "Categories for the Working Mathematician" by Saunders Mac Lane, chapter IV.3, p.93. In any category $C$ a skeleton of $C$ is any full subcategory $A$ such that each object of $C$ is ...
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3 votes
1 answer
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Spherical fusion categories: A certain functor

1. Context Let $C$ be a spherical fusion category over an algebraically closed field $k$ of characteristic zero. Denote by $Vec$ the category of finite-dimensional vector spaces. Currently, I am ...
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4 votes
2 answers
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If $A, B$ are Morita equivalent, then $\underline{\text{Nat}}(I_A, I_A)\cong\underline{\text{Nat}}(I_B, I_B)$.

Let $A, B$ two rings and $I_A: {}_A\text{Mod} \to{}_A\text{Mod}$ the identity functor. I am trying to show that if $A, B$ are Morita equivalent, then $\underline{\text{Nat}}(I_A, I_A)\cong\underline{\...
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0 votes
1 answer
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A natural transformation

On the page $9$ here I have a very basic question: in the definition $3.1$ $\alpha$ never appears in the condition "such that if $(U_1f)a\in U_2 HB$ then $a\in U_2 HA$ for each $f:A\to B$ in ${\...
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4 votes
2 answers
262 views

Equivalence Between Category of Covering Spaces and Category of Sets with an Action By The Fundamental Groupoid

Let $\Pi_1(X)$ be the fundamental groupoid of a locally path-connected topological space $X$ and define $\Pi_1(X)-\mathbf{Sets}$ to be the category of sets equipped with an action by the fundamental ...
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  • 460
0 votes
2 answers
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are two the following functors isomorphic

Consider the following two functors that go $C^{op}\times C\to Set$, where $X$ is an object in $C$ $(A,B)\mapsto Set(C(X,A)\times A,B)$ $(A,B)\mapsto Set(C(X,A),Set(A,B))$ Are these two functors ...
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5 votes
1 answer
197 views

Functor From Category of Covering Spaces to Category of Sets Equipped With An Action By The Fundamental Groupoid

I have some problems with the understanding of the connection between covering spaces and the fundamental groupoid. Let $X$ be a topological space and let $\Pi_1(X)$ denote the fundamental groupoid of ...
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5 votes
1 answer
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Reconstructions of Groups From Category of $G-\mathbf{Sets}$; Construction of a Group Homomorphism [duplicate]

I try to come up with a proof of the following statement, but I find it a little difficult. I hope that I can get some help from someone on this site. I think this is what they give a proof of, on ...
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1 vote
0 answers
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Naturality conditions in the definition of adjoint operation

Two functors $F: C \mapsto D$ and $G: D \mapsto C$ are said to be adjoint if there exists a natural bijection $\tau_{A,B}: Mor(F(A), B) \mapsto Mor(A, G(B)) \quad \forall A \in C \; \forall B\in D.$ I ...
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2 votes
0 answers
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Composition structure on ${\bf Fun}$

Let Fun denote the category of functors and natural transformations. Does composition of functors together with the Godement product of natural transformations amount to some sort of canonical ...
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2 votes
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Natural transformation $\mathbb{A}^n \setminus \{ 0 \} \to X$.

Let $\text{Ring}$ the category of commutative ring and $\text{Set}$ the category of set. Denote by $\Omega : \text{Ring} \to \text{Set}$, the functor $R \mapsto \{ \text{Ideal of $R$} \}$, for a ring ...
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1 vote
1 answer
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What Does it Mean for a Splitting to be Natural?

$\require{AMScd}$ I'm reading through the proof of the Universal Coefficient Theorem (for homology) given in Massey's Singular Homology Theory, in which he claims that the split SES $$0 \to H_n(K) \...
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1 vote
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Proving the Naturality of the Tor Functor

$\require{AMScd}$ I've been reading about the Ext and Tor functors from a number of sources which all claim that the Tor functor provides a natural way of extending the exact sequence of abelian ...
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4 votes
0 answers
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Horizontal composition of pseudonatural transformations

Let $A$, $B$, and $C$ be 2-categories (or bicategories, etc), and let $F,F':A\to B$ and $G,G':B\to C$ be 2-functors (or pseudofunctors, etc). Now let $\alpha:F\Rightarrow F'$ and $\beta:G\Rightarrow G'...
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  • 7,397
4 votes
1 answer
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Natural transformations $Id_{Ab} \rightarrow Id_{Ab}$

Let $Id:\frak{Ab} \rightarrow {Ab}$ be the identity functor of $\frak{Ab}$ (category of abelian groups). The class of natural transformations $\eta: Id \rightarrow Id$ is a monoid under operation ...
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  • 336
2 votes
0 answers
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Every representable presheaf is projective

Let $\mathcal{C}$ be a small category and let $\hat{\mathcal{C}}$ be its category of presheaves. I want to show that every representable presheaf $y_C\in \hat{\mathcal{C}}$ (for some $C\in\mathcal{C}$)...
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