# Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

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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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### 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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### 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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### 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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### 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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### 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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### Natural transform and Fourier transform?

You can generalize the Fourier transform to a functor F from the space of tempered distributions onto itself. Do we know some interesting natural transformations from F or to F? It is likewise o. k. ...
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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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### 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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### Are equivalence of triangulated categories exact? [duplicate]

Let $D, E$ be triangulated categories and for my setting I take the shift functors to be auto-equivalence, and let me use the same notation $\Sigma$ for the shift. Let $F: D \to E$ be an equivalence ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### Categorical interpretation of modules (over algebras)

Let $k$ be a field and $F_n$ be a finitely generated $k$-algebra with $n$ generators. Then, a $F_n$-module is the same thing (more precisely, there is an isomorphism of categories) as a $k$-vector ...
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### Kunneth formula proof doubt, and natural transformation

I have been reading MacLane's Book on Homology and I have a doubt in the proof of the Kunneth Formula In $10.5)$ he he says that $\bigoplus D_{m+1}\otimes H_q(L) \cong H_{n+1}(D \otimes L)$, and i ...
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### $C_n(X;G)$ is naturally isomorphic to $C_n(X) \otimes G$
Let $X$ be a space, and let $G$ be a fixed group. What does "$C_n(X;G)$ is naturally isomorphic to $C_n(X) \otimes G$" means? I know that these two groups are isomorphic, since the following hold: \$...