Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

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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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Natural homomorphism involved in the Kunneth-Formula

I have been reading the proof of the Kunneth-Formula in MacLane's Homology book, and to prove that is natural he does this, $(10.6)$ (sorry for posting a picture but I dont know how to diagrams ...
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1answer
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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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2answers
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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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Limits and colimits chasing: why $h$ is iso? [duplicate]

I do not follow why $h$ in the second snippet below is iso from the fact that $$p=q.$$ Why is it injective and why surjective ?
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1answer
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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 ...
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2answers
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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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1answer
71 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: $...
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1answer
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Natural transformations as categorical homotopies

I was reading this question with the same title at MathOverflow, which defines natural transformations in the following way: given two functors $\mathcal F,\mathcal G \colon \mathcal C \to \mathcal ...
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1answer
49 views

Adjoints of two naturally isomorphic functors

Suppose that $(F, G)$ and $(F', G')$ are two pairs of adjoint functors, and moreover that $F$ and $F'$ are two naturally isomorphic functors. It is true that $G$ and $G'$ are also naturally ...
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1answer
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Describing a natural transformation in terms of hom-sets.

I was reading "Categories for the Working Mathematician" by Saunders Mac Lane and on page 28 I was puzzled by a question. Hom-sets were being explained and the author leaves it to the reader to ...
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1answer
85 views

Cisinski, Higher Categories and Homotopical Algebra, Theorem 1.1.10

Let $X\colon \mathsf{C^{op}}\to \mathsf{Set}$ be a presheaf. It's category of elements, denoted by $\int X$, has pairs $(a,s)$ with $s \in X(a)$ as objects and $f \in \mathrm{Hom}_{\mathsf{D}}(a,b)$ ...
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Naturality of certain isomorphisms for $\mathrm{Hom}$-functors

Apparently, it is known that there are isomorphisms $$\mathrm{Hom}_R(A,\prod_{i \in I} B_i)\cong\prod_{i \in I} \mathrm{Hom}_R(A, B_i)$$ and $$\mathrm{Hom}_R(\bigoplus_{i \in I} A_i, B)\cong\prod_{i \...
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1answer
46 views

Not all Monads are Idempotent, a Cautionary Tale on Natural Transformations

I was scratching my head over the following for almost an hour today, and since I don't have a blog to share the resolution on, I'll post it here. Let $\mathbb{T} = (T,\eta,\mu)$ be a monad on a ...
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Riehl's Category Theory in Context - Exercise 1.5.vii without Axiom of Choice

From Emily Riehl, Category theory in context: Exercise 1.5.vii. Let $\mathbf{\mathsf G}$ be a connected groupoid and let $G$ be the group of automorphisms at any of its objects. The inclusion $\...
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1answer
44 views

Extension of Scalars is a pseudofunctor

Let $\boldsymbol{Ring}$ denote the category of commutative rings with unity (not necessarily different from $0$), and morphisms preserving unity. I have been wondering if the assignment $A\in\...
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1answer
66 views

Trouble understanding what a natural transformation is [duplicate]

I understand that an arrow is between two objects, a functor is between two categories. And then a natural transformation is, according to Goldblatt's Topoi, a comparison of two functors. Here is ...
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Natural transformation induced by adjoint functors. [duplicate]

My question is the same as here, but I can't understand why the transformation from the identity functor to the composition of the adjoint functors is natural. I've tried proving explicitly that the ...
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Why unnatural isomorphisms are not mathematically well- behaved?

It’s said in nlab that Unnatural isomorphisms are not very well-behaved mathematically Could you provide some examples illustrating this ? Thanks for any suggestion.
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Dinatural isomorphism from tangent functor to cotangent functor

It is well known that the identity and dual space functors on the category $\mathrm{Vec}_\mathbb{R}$ are not naturally isomorphic because one is covariant and the other contravariant. We can then ...
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The Grassmannian of a quasi-coherent module

In Goetz' and Wedhorn's Algebraic Geometry Chap (8.6) page 214 is introduced the concept of generalized Grassmannian's. One remark I can't understand. (8.6) The Grassmannian of a quasi-coherent ...
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1answer
68 views

Proving that the horizontal composition of natural transformations is a natural transformation

Is there an easy way to prove that the horizontal composition of natural transformations is a natural transformation? To fix the notation, here is part of Leinster's textbook: Suppose $f:A\to A'$ is ...
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1answer
55 views

Promoting equivalence to adjoint equivalence: explicit construction of a unit and a counit

Let $F\colon\mathsf{C}\to\mathsf{D}, G\colon\mathsf{D}\to\mathsf{C}$ together with $\eta\colon 1_{\mathsf{C}}\to \mathsf{GF}, \epsilon\colon\mathsf{FG}\to 1_{\mathsf{D}}$ be an equivalence of ...
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1answer
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Bijection between $\textbf{Cat}(A \times B, C)$ and $\textbf{Cat}(A, C^B)$: Mac Lane Exercise 1 Chapter II section 5

I'm experiencing some confusion regarding the following exercise in Chapter 2, Section 5 from Mac Lane's Categories for the working mathematician: For small categories $A$, $B$ and $C$ establish a ...
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1answer
39 views

Clarification on Natural Transformations of Bifunctors

I'm having a hard time understanding the following excerpt which appears on page 38 of Mac Lane's Categories for the Working Mathematician: Next consider natural transformations between bifunctors $...
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1answer
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modification of $2$ natural transformations

I'm reading this part of Borceux Handbook of Categorical Algebra, and I have a problem with the equation on the last but one line in the snippet: $$\Xi_{A'}\ast F\alpha=G\alpha\ast \Xi_A.$$ My ...
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1answer
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Natural isomorphism between the squaring functor and the identity functor on infinite sets

Let $C$ be the full subcategory of the category of sets consisting of the infinite sets. Is the endofunctor $X \mapsto X \times X$ on $C$ naturally isomorphic to the identity functor? We know that ...
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1answer
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What does it mean for an isomorphism to depend on a basis?

In an attempt to understand dual spaces and adjoints (in linear algebra) I came accross this video, which mentions "natural isomorphisms". Not knowing what a natural isomorphism is, I tried to look it ...
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2answers
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Naturality vs canonicality of double dual

$\require{AMScd}$ I'm learning some linear algebra and basic category theory, and have been asked to show two things. There is a canonical map $f: V \rightarrow V^{**}$ The following diagram commutes:...
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Reformulation of equivalence of categories using natural isomorphisms

What is the proof of equivalence of these two conditions for dummies? (1) $F:\cal K\to L$ is full, faithful and essentially surjective on objects (2) $\cal K$ and $\cal L$ are equivalent, i.e. the ...
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About the naturality condition in the Yoneda lemma

After reading the statement of the Yoneda lemma (Theorem 4.2.1), I understand that it states that the functor $[\mathscr A^{op},\mathbf{Set}](H_\bullet,-)$ is naturally isomorphic to the functor $-(\...
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1answer
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Naturality in two variables is equivalent to naturality in each variable

Exercise from Leinster: I managed to prove one direction as follows. Suppose the mentioned family is a natural transformation $F\implies G$. Then for all arrows $f:A\to A'$ and $g:B\to B'$, $$\...
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1answer
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Is there always a natural transformation from one function to another; and is it unique?

Categories for the Working Mathematician says Given two functors $S, T: C \to B$, a natural transformation $r: S \to\to T$ is a function which assigns to each object $c$ of $C$ an ...
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1answer
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On the notion of “naturally in” (within the context of natural isomorphisms)

I currently work through Tom Leinster's Basic Category Theory and I am not sure if I really grasp the notion of naturally in $A$. Since this sentence does not make any sense stated isolated I will ...
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1answer
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2-functor and CAT

Let $\cal K$ be a small category. Let $\cal A$ be a subcategory of $\mathbf {CAT}$ and $U:{\cal A}\hookrightarrow{\mathbf {CAT}}$ the underlying functor. Now how is $U^{\cal K}$ naturally defined as a ...
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Composition of a natural transformation with a functor

I have asked a similar question elsewhere but still in the snippet below, I don't know what is the functor $G$ and how differs $(G\ast\omega_F)$ from $(\omega_G\ast F)$ , what are they and what is $\...
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Kan Extensions and Coends

I am lost in a bit of a haze of abstraction and am wondering if someone can set me straight. I'm reading the chapter in Categories for the Working Mathematician, and Mac Lane points out on page 238 ...
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1answer
101 views

Composition of a natural transformations with a functor--whiskering [duplicate]

If $M$ is an endofunctor on a category $\cal K$ and $\eta:Id_{\cal K}\to M$ is a natural transorfmation, what is the difference between $\eta M$ and $M\eta$, and how these two (componentwise) are ...
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Is $H_n(X,A)$ naturally isomorphic to $\tilde{H}_n(X/A)$?

It is well known that under some assumptions on a pair $(X,A)$ of a topological space and a subspace we have $H_n(X,A)\simeq \tilde{H}_n(X/A)$. Such an assumption can be for example that $A$ is closed ...
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1answer
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$\Theta$ is induced by the inclusion map $Z_p\otimes Z_q' \to (C\otimes C')_{p+q}$?

Let $C$ and $C'$ be two chain complexes and $C\otimes C'$ their tensor product. Then define the map \begin{align} \Theta : H_p(C)\otimes H_q(C')&\to H_{p+q}(C\otimes C')\\ {[z_p]}\otimes {[z_q']}&...
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1answer
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Splitting induces isomorphisms

Let $M^n$ be a closed topological manifold. For $1 \leq q \leq n$, we have the following short exact sequence: $$ 0 \longrightarrow \operatorname{Ext}(H_{q-1}(M),\mathbb{Z}) \stackrel{\beta}{\...
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1answer
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Natural transformation between functors $\mathbf {CRing}\to\mathbf{Mon}$

Regarding Example 1.3.5: 1) How to see that a ring homomorphism $R\to S$ induces a monoid homomorphism $M_n(R)\to M_n(S)$? It definitely induces a map, but why is it a monoid homomorphism? 2) Why do ...
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What structure do natural isomorphisms preserve?

My understanding from model theory is that, given groups A and B, the statement $ A \cong B $ implies that any for any first order statement $P$ in the language of groups, $P(A) \iff P(B) $. Can an ...
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1answer
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Naturality of an exact sequence

I was reading this post about the Künneth theorem, to try to understand what it is. I don't really understand what it means for a sequence to be natural. Does this mean we have a natural ...