Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

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Topological analogue to natural transformation? (in the same way that natural isomorphisms are categorical homotopies)

Let $X,Y$ be topological spaces, $\mathsf{C},\mathsf{D}$ categories, $I$ the interval, $\mathbb{2}$ the 2-object category with one nontrivial morphism, and $\mathsf{I}$ the 2-object category with two ...
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Question on why the isomorphism $A \cong \mathbf{Z}^n \oplus \text{Tor} A$ is not natural -- A clarification of Riehl's choice of group?

I'm reading Category Theory in Context, and I have a clarification question. Her Proposition 1.4.4 says that the isomorphism of f.g. Abelian groups $A \cong \mathbf{Z}^n \oplus \text{Tor} A$ is not ...
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Colimit of categories don't preserve equivalences [duplicate]

I have a question about the following problem: Find two diagrams of shape J in Cat $F,G:J\rightarrow Cat$ and a natural transformation $\eta:F\rightarrow G$ such that $\eta_i:F(i)\rightarrow G(i)$ is ...
1 vote
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77 views

$F\dashv G\dashv F\implies F,G$ are equivalences of categories

Let $F:C\to D$, $G:D\to C$ be functors such that $F\dashv G\dashv F$. I want to show that they are equivalencies of categories.We have the existence of $\eta:Id_C\to G\circ F$, $\epsilon:F\circ G\to ...
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1 vote
1 answer
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Equivalence of categories and colimits

Consider functors $F,G:J\to Cat$ which admit colimits and $\eta:F\to G$ a natural transformation such that $\forall i\in J$ $\eta_i:F(i)\to G(i)$ is an equivalence of categories. Is it true that then ...
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Category of functors with horizontal composition

Consider $\epsilon:F\to G$, $\mu:K\to L$ to be natural transformations between functors $F,G:C\to D$, $K,L:D\to E$. We can define a (horizontal) composition law by $(\mu*\epsilon)(c)=L(\epsilon_c)\...
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1 answer
44 views

Show naturality of $\infty$-natural transformation

Working with the model category of complete Segal spaces $\text{CSS}$, which has as its underlying category the category of simplicial presheaves on $\Delta$, one has a suitable internal hom in $\text{...
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1 answer
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What is the difference between functor composition and natural transformation?

If $F$ and $G$ are injective functors between the categories $C$ and $D$ $H$ is an endofunctor on the category $D$ such that $H∘F = G$ $η$ is a natural transformation from $F$ to $G$ Then for every ...
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Definitions of adjoint functors

Definition. Let $F:\mathcal A\rightleftarrows\mathcal B:G$ be a pair of functors. We say that $F$ is left adjoint to $G$ and write $F\dashv G$ if there are exists natural transformations $\varepsilon:...
4 votes
1 answer
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Uniform Choice Functions and Naturality

Some choice functions can be specified explicitly, while in other cases no definite choice function is known. An example of the former is a choice function for non-empty subsets of natural numbers, ...
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2 answers
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Composition of morphism part of evaluation (bi)functor.

Before giving a lenghty introduction, I'd like to actually just ask one thing. We are given the object part Ev$_0$ of the evaluation functor $\mathcal{C} × [\mathcal{C}, \mathcal{D}] → \mathcal{D}$. I'...
1 vote
2 answers
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Alternative functor construction from universal morphisms

Let $G : \mathcal{D} \rightarrow \mathcal{C}$ be a functor. Suppose that for each object $X \in \mathcal{C}$, there exists a universal morphism $(F_X, \eta_X)$ from $X$ to $G$. The theory of adjoint ...
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2 votes
1 answer
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Why are natural isomorphisms injective on objects?

Here, here, and here says a natural isomorphism $\eta \colon F \rightarrow G$ can be regarded as a natural transformation with a two sided inverse, or alternatively each $\eta_X$ is an isomorphism. ...
1 vote
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How to check a morphism is actually a morphism between functors needed to construct an adjoint characteristic

I am learning elementary category theory. My question was raised while I was reading a part about a characteristic of adjoint functors. We adopt a policy for the notation $h_A(X) = \mathrm{Mor}(X, A)$ ...
2 votes
1 answer
64 views

In the axioms of a coalgebra, does the *naturalness* of the isomorphisms play any role?

I don't know whether this question makes complete sense, but I'm 90% certain it does. In the definition of a coalgebra over a field, the fact that $(C \otimes C) \otimes C \cong C \otimes (C \otimes C)...
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1 answer
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What is variable substitution best thought of categorically? A natural transformation?

Here is an attempted proof in the category $\textbf{Ass}$ where objects are assertions (in a kind of ordered-and or CNF form - essentially a list of assertions) and morphisms are "proofs" ...
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1 answer
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Describing epimorphisms in Set-valued functor categories without using pointwise computation of colimits

Let $\mathscr{A}$ be a small category and consider the functor category $[\mathscr{A}, \mathbf{Set}]$. Fact. The epimorphisms in $[\mathscr{A}, \mathbf{Set}]$ are precisely those natural ...
3 votes
1 answer
75 views

All natural transformations from $\operatorname{id}_{\textsf{Grp}}$ to itself

I am trying to find all natural transformations from the functor $\operatorname{id}_{\textsf{Grp}}$ to itself. $\require{AMScd}$ \begin{CD} G @>{\phi}>> H \newline @V{\alpha_G}VV @VV{\alpha_H}...
3 votes
1 answer
96 views

Why the triangle diagram commutes because of naturality

I am new in category and I am reading Awodey's Category Theory. In the proof of Proposition 8.10 (See the picture of the proof here1, here2 and here3), after identifying $$x\in P(C)$$and $$x:yC\...
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Natural Isomorphism $(Y^Z)^X\cong Y^{Z\times X}$ in a cartesian closed category

Let $\mathcal{C}$ be a cartesian closed category. I'm working on a problem that asks me to show that for $X,Y,Z\in\text{ob}(\mathcal{C})$ there is a natural isomorphism $(Y^Z)^X\cong Y^{Z\times X}$. ...
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1 answer
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Is there a 'coherence' theorem for simplicial sets? On uniquely lifting natural transformations of $n$-truncations to full natural transformations

$\newcommand{\set}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\newcommand{\sk}{\operatorname{sk}}\newcommand{\tr}{\operatorname{tr}}\newcommand{\cosk}{\operatorname{cosk}}\newcommand{\nat}{\mathsf{...
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1 vote
1 answer
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Ideal-vanishing locus correspondence in affine schemes as natural transformations

$\DeclareMathOperator{\Spec}{Spec}$$\newcommand{\pf}{\mathfrak{p}}$Let $A$ be a ring (commutative, with $1$), then its spectrum $\Spec A$ is the set of prime ideals of $A$. It can be turned into a ...
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Theorem about horizontal composition [duplicate]

I want to prove Theorem 1 in section 2.5 of McLane's Categories for the Working Mathematician, which was left to the reader. The theorem: The collection of all ...
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1 answer
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Is it true that functors which are surjective on objects are obviously essentially surjective?

I am asking this as I have established a functor F between categories C and D such that F is faithful, full and surjective on objects. Can I say that F is an equivalence of categories? I think so but ...
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Isomorphic vector bundles have same characteristic classes

Following is an excerpt from the book "Differential Geometry: Connections, Curvature, and Characteristic Classes" that defines naturality of characteristic classes. My question is how does ...
3 votes
1 answer
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Understanding Marsden's string diagrams for naturality

I'm trying to learn about string diagrams from Dan Marsden's tutorial, and I first get confused on page 9, when he introduces the following two diagrams and claims that they express the naturality ...
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2 votes
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Universal property of the (Vistoli-)sheafification

Given a presheaf $F$, in Notes on Grothendieck topologies, fibered categories and descent theory there is a construction of the sheafification (Proof for theorem 2.64). The first part is the ...
1 vote
1 answer
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How to understand in context of category theory: combination of contravariant functor and natural transformation

I have a category $\mathcal{C}$, a contravariant functor $D:\mathcal{C}\to\mathcal{C}$ (so that $\text{id}_{\mathcal{C}}$ and $D^2$ are covariant), and a natural transformation $\chi:\text{id}_{\...
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2 votes
1 answer
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Terminology: contravariant functor $F:\mathcal{C}\to\mathcal{C}$ for which $F^2$ is naturally isomorphic to the identity

I have a category $\mathcal{C}$ and a contravariant functor $F:\mathcal{C}\to\mathcal{C}$, so that $F^2$ and $\text{id}_{\mathcal{C}}$ are covariant functors $\mathcal{C}\to\mathcal{C}$. I also have a ...
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2 votes
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Extending natural transformation from a set of projective generators

$\mathcal{C}$, $\mathcal{D}$ be abelian categories. $\mathcal{C}$ is cocomplete, and has set $\mathcal{P}$ of projective generators. $F$, $G\colon \mathcal{C} \to \mathcal{D}$ are additive functors ...
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2 votes
1 answer
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Natural Transformation and the freedom from artificial choices

I'm reading Mac Lane's "Categories for the Working Mathematician" On section $I.4$ on Natural Transformation, I've ran into into a statement I could not wrap my head around. Given an abelian ...
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1 answer
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Is flattening a natural isomorphism?

Consider the following categories: $𝕍≔\operatorname{FinVec}_ℝ$ of finite dimensional real vector spaces, whose morphisms are the linear maps $ℍ≔\operatorname{FinHilb}_ℝ$ the category of finite ...
2 votes
1 answer
132 views

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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4 votes
0 answers
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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), ...
2 votes
1 answer
181 views

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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1 vote
1 answer
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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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2 votes
0 answers
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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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1 answer
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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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1 answer
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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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0 votes
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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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2 votes
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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 $C_1, C_2 \in Ob(\mathcal{C})$, $F(C_1) = F(C_2) \implies G(C_1) = G(C_2)$, ...
3 votes
1 answer
156 views

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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1 answer
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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\...
0 votes
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55 views

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 ...
3 votes
1 answer
169 views

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_\...
1 vote
0 answers
74 views

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}{\...
1 vote
1 answer
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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 $$H_{k}(X) \approx \sum_{\alpha \in A}H_{k}(X_{\alpha}).$$ ...
1 vote
1 answer
139 views

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. ...
1 vote
1 answer
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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 ...
1 vote
1 answer
91 views

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(\...