# 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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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... 0 votes 0 answers 44 views ### 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 ... 3 votes 1 answer 63 views ### 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 ... 0 votes 0 answers 31 views ### 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 ... 2 votes 1 answer 41 views ### 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 ... 1 vote 1 answer 79 views ### 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 ... 2 votes 1 answer 47 views ### 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}^\... 0 votes 1 answer 53 views ### 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: ... 0 votes 1 answer 91 views ### 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) ... 1 vote 2 answers 79 views ### 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/... 2 votes 1 answer 43 views ### 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 ... 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 ... 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 ... 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 ... 7 votes 1 answer 201 views ### 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}] ... 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 ... 1 vote 1 answer 47 views ### 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 ... 3 votes 1 answer 103 views ### 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 ... 4 votes 2 answers 73 views ### 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{\... 0 votes 1 answer 42 views ### 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 {\... 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 ... 0 votes 2 answers 51 views ### 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 ... 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 ... 5 votes 1 answer 205 views ### 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 ... 1 vote 0 answers 23 views ### 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 ... 2 votes 0 answers 67 views ### 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 ... 2 votes 0 answers 69 views ### 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 ... 1 vote 1 answer 142 views ### 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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$\require{AMScd}$ 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 ...