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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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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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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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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Natural transformations and 2-cells

I have a strict $2$-category $\mathcal{C}$ having $2$-cells $\sigma : gh \Longrightarrow fh$, and $\delta : h \Longrightarrow h^2$, where $f : A \longrightarrow B$, $g : A \longrightarrow B$ and $h: A ...
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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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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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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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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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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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When is there an “intuitive” functor from F-coalgebras to T-coalgebras?

Suppose $F, T : Set \rightarrow Set$ are two functors on the category of sets. Let $F^{coalg}$, $T^{coalg}$ denote the categories of $F$, respectively $T$ coalgebras. Vaguely, I'm interested in when ...
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102 views

Inductive limits commute “naturally” with binary products in Set

I am taking an introductory course in Category Theory, and one of the problems is Prove that inductive limits commute with binary products in Set; i.e. for infinite sequences of sets $\{X_n\}_{n \in \...
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1answer
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Property of 2-natural transformations

Let $C$ and $D$ be (strict) 2-categories, let $F,G:C\to D$ be (strict) 2-functors, and let $\alpha:F\Rightarrow G$ be a (strict) 2-natural transformation. Let now $X$ and $Y$ be objects of $C$, let $f,...
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use naturality to show bimodule homomorphism

Let $A, B$ be $k$-algebras, and let $M, M'$ be $A-B$-bimodules. There is a bijection $Nat(Hom_A(M, −), Hom_A(M', −)) \cong Hom_{A⊗_kB^{op}} (M', M)$ sending a natural transformation $η : Hom_A(M, −) \...
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closed, connected, orientable manifold with nonzero induced homology maps $S^n \rightarrow M$. Compute $H_k(M;\mathbb{Q})$

Let $M$ be a closed, connected, orientable $n$-dimensional manifold, and suppose there is a map $f:S^n\rightarrow M$ such that the induced homomorphism $f_*:H_n(S^n;\mathbb{Z})\rightarrow H_n(M;\...
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1answer
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Naturality of $\varphi : \textbf{Vct}_K(V(x), w) \xrightarrow{\sim} \textbf{Set}(x, U(w))$ in the variable $x$ (Cats for the Working Mathematician).

Consider the forgetful functor $U : \textbf{Vct}_K \to \textbf{Set}$ and the functor $V : \textbf{Set} \to \textbf{Vct}_K$ that takes an object $x$ in set to the $K$-vector space $V(x)$ with basis $x$ ...
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Why is important to know if transformations between functors is natural or not?

I'm studying category theory and I understand what a natural transformation is. I also understand that intuitively speaking it's like defining a polymorphic function that's independent of types, so to ...
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Using Yoneda to establish natural isomorphisms?

I know the Yoneda embedding can be used to easily establish isomorphisms between objects in categories. For example, in a locally small cartesian closed category $\mathbf{C}$ with coproducts, the &...
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Does isomorphism of sets of natural transformations imply isomorphism between functors?

I am studying Proposition 3.7.4 of Francis Borceux's Handbook of Categorical Algebra. For specific functors $F,G:\mathcal{B}\rightarrow\mathcal{D}$, he shows that for any functor $H:\mathcal{B}\...
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A question about adjoint functors:

I am trying to replicate the solution to this question. But, in the process I got stuck. Here is my question: Suppose $F: C\rightleftarrows D :G$ is an adjoin pair with unit $\eta :1_C\Rightarrow GF$ ...
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“Natural” equivalence of categories?

Let $\mathbf{C}$ be a category and $F,G:\mathbf{C}\to\mathbf{Cat}$ be category-valued functors on $\mathbf{C}$. Suppose there is a family of equivalences of categories $$(\Phi_C:FC\simeq GC)_{C\in\...
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Monoidal adjunction whose right-adjoint functor has structure morphisms which are epimorphisms

Let $(\mathbf{C},\otimes,1)$ and $(\mathbf{D},*,e)$ be monoidal categories and let $L:\mathbf{C}\rightarrow \mathbf{D}$ and $R:\mathbf{D}\rightarrow \mathbf{C}$ be functors. Suppose that there exists ...
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Generalized diagrams

Certain mathematical objects can be described as a functor $F : \mathcal{J} \to \mathcal{C}$ from a small index category $\mathcal{J}$ to a bigger category $\mathcal{C}$. For example, we can think of ...
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1answer
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Natural isomorphism in linearly distributive categories with left and right dualities

Let $\mathrm{C}$ be a linearly distributive category with a left and right duality, i.e. it is a monodical category "twice": once for the bifunctor $\otimes:\mathrm{C}\times\mathrm{C}\to\mathrm{C}$ ...
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1answer
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The inverse of a natural isomorphism is a natural isomorphism

Let $F,G:\mathcal{C}\to \mathcal{D}$ be functors and let $\alpha:F\Rightarrow G$ be a natural transformation between them. Suppose that, for every object $C\in\mathcal{C}$, the morphism $\alpha_C:FC\...
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1answer
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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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Various examples of non-representable functors

A (covariant) functor $F:\mathbf{C}\to\textbf{Set}$ with domain a locally small category $\mathbf{C}$ is said to be representable if it is naturally isomorphic to the hom functor $\text{Hom}_{\mathbf{...
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Natural transformation between $\mathcal{F}$ and $\mathcal{G}$

Consider the following functors from the category of finite Abelian groups $F AbGrps$ to the category of sets defined as follows: $\mathcal{F}: F AbGrps \to Sets$, a group $G$ is mapped to the set ...
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Self-Natural Maps for Forgetful Functor

Let $\mathscr{F}$ denote the forgetful functor from the category of groups to the category of sets. Why is there more then one natural map from $\mathscr{F}$ to $\mathscr{F}$? What are all of the ...
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Can the left/right-inverse of natural transformation be analogous to the left/right inverse of a function between hom-set?

Let's look at the right inverse of a function first: Let $f:X→Y$, $g:Y→X$ is right-inverse of $f$ (or section of $f$), if only if , $f∘g=id_{Y}.$ It means that If $X$ and $Y$ are finite sets, then $|...
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Question concerning the existence of natural transformations of two conjugate functors of groups regarded as categories.

I am reading and learning some basic category theory. I found the following problem on Categories for the Working Mathematician (I understand that it may not be the best learning source for a beginner,...
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73 views

$C_n(X;G)$ 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: $...