Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

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Among morphisms of morphisms, what makes commutative squares special?

Given two (1-)categories $\mathcal{C}, \mathcal{D}$, and given the 0-category (class) of funtors $\mathcal{C} \to \mathcal{D}$, denoted $Func(\mathcal{C} \to \mathcal{D})$, let's say we want to make ...
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Proof and explanation of a vector space not being naturally isomorphic to its dual

From https://ncatlab.org/nlab/files/SelingerSelfDual.pdf It is well-known that each finite dimensional vector space A is isomorphic, but not naturally isomorphic, to its dual space A∗. I would like ...
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Left and right adjoints of functor category inclusion

Question. Let $i:\mathcal{C}\hookrightarrow\mathcal{D}$ be a full subcategory. Assume we are given a cocomplete category $\mathcal{A}$. Show that the induced pre-composition functor $i^*:\mathrm{Fun}(\...
Robert's user avatar
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Why are the isomorphisms induced by the category with finite products natural?

A follow-up question to this. Source: Categories for the Working Mathematician, second edition by Saunders Mac Lane. Proposition: If a category $C$ has a terminal object $t$ and a product diagram $a\...
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What are the natural transformations induced by a category with finite products?

Source: Categories for the Working Mathematician, second edition by Saunders Mac Lane. Proposition: If a category $C$ has a terminal object $t$ and a product diagram $a\leftarrow a\times b\rightarrow ...
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What exactly is the natural transformation that arises in a coproduct diagram?

Source: Categories for the Working Mathematician, second edition by Saunders Mac Lane. Given a category $C$, a coproduct diagram is a universal arrow from an object $<a, b>$ of $C\times C$ to ...
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Fibered category with cleavage is a pseudofunctor

I'm following Vistoli's note on fibered categories and stacks. There Proposition 3.11 he states, which as I understand is a classical result with many references, which says $-$ A fibered category ...
stratified's user avatar
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Yoneda lemma for representable profunctors?

For $F:C\rightarrow D$ a functor, consider the profunctor $h_F:=\operatorname{Hom}_D(-,F-):D^{op}\times C \rightarrow \mathsf{Set}$. For a functor $G:C\rightarrow D$, define $h_G$ similarly. Is it ...
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Proving naturality in both arguments

I am having a hard time showing that two functors are adjoint. Consider as an example \begin{align} t&: \mathbf{Ab} \longrightarrow \mathbf{T},\\ i&: \mathbf{T} \longrightarrow \mathbf{AB}. \...
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Question on adjoints and natural bijections (Altman & Kleiman Exercise 6.14)

This is Exercise 6.14 from A Term of Commutative Algebra by Altman and Kleiman. Let $\mathcal{C}$ and $\mathcal{C}'$ be categories, $F:\mathcal{C}\to\mathcal{C}'$ and $F':\mathcal{C}'\to\mathcal{C}$ ...
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Proving the existance and uniqueness of the unit of an adjunction

Let $L: C \to D$ be left adjoint to the functor $R: D \to C.$ For an object $X$ of $C$, we have an isomorphism $Hom_D(L(X),L(X)) \xrightarrow{\sim} Hom_C(X,(R\circ L)X)$. Show that there exists a ...
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About the double dual of Lipschitz functions

I am referring to the introduction of this paper: G. Flores - On an identification of the Lipschitz-free spaces of $\mathbb{R}^n$ Here, the authors describe the predual space of the space $$ Lip_0(M) :...
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'Uniqueness' of adjoint functors

Let $F:\mathcal{A}\to\mathcal{B}$ be a left-adjoint functor. What 'choices' need to be made in order to construct a right-adjoint $G:\mathcal{B}\to\mathcal{A}$ for $F$ and a natural isomorphism $$\...
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A "natural transformation" in which the "morphisms" are not morphisms

$\newcommand{\G}{\mathcal G}$I have been playing around with the category of groups, and seeing if it can be defined in a purely category-theoretic way (without appeal to binary operations and inner ...
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Natural transformations and canonical maps

I’ve had a lingering question that I haven’t been able to fully resolve. I’ve often noticed that many demonstrations don’t detail the proof of natural transformation, instead simply stating that the ...
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Extranatural transformations are a special case of dinatural transformations

Recently I have been trying to understand extranatural and dinatural transformations. Here are the definitions I am working with. From the nLab: Let $F:A\times B\times B^{op}\to D$ and $G:A\times C\...
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Defintion of relative cohomology cross product

This is a passage from my notes which lacks some details: Let $X,Y$ be topological spaces, and $A\subseteq X$ and $B\subseteq Y$ be open subsets. Then there is a natural map $K\colon S_{\ast}(X,A)\...
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How many universal cones from $\Delta_{A\times A}$ to D?

I have drawn the picture like the following. Let $C$ is a category (suppose $Set$), $I$ is a category with only two objects (1 and 2), $\Delta_{A\times A}$ and $D$ are constant functors from $I$ to C....
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$\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ on finite abelian groups is not right exact?

In his answer here Martin Brandenburg claims that the Tor functor $\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ in the category of finite abelian groups is not right exact in neither argument. Since Tor is ...
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Conjugation Functor from a Groupoid to $\mathbf{Grp}$

Take a groupoid $\mathcal{C} \in \mathbf{Grpd}$. It's possible to construct a conjugation functor $F_{\mathcal{C} } : \mathcal{C} \to \mathbf{Grp}$ as follows: For every object $x \in \text{ob}(\...
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What are colored morphisms/arrows intended to mean in these diagrams?

I've been reading more category theory as a prerequisite to understanding some more complicated theorems, and for the first time I'm running into arrows that are distinctly colored. Examples include ...
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Natural isomorphism between the complexification of $V$ and $V^2$, when $V$ is already a complex vector space?

I'm considering the question of: What happens if you complexify a vector space that is already complex? I basically believe the following. I take complexification of a complex vector space to mean ...
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Understanding the solution of Exercise 4.1.32 in Tom Leinster "Basic Category Theory".

Here is the exercise and its solution: 1-Is there a typo and $\varphi$ should be $\psi$? 2- I do not understand how by exercise 2.1.14 we will get the first equation in the solution of 4.1.32. And ...
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Do all Functors have a natural transformation from the identity functor? [closed]

Considering categories C and D, and functor $F:C\to D$, and the identity functor $Id_C : C\to C$. Is there always a natural transformation $n: Id_C \to F$, where n(X) is just F(X)? $$\require{AMScd} \...
segfault's user avatar
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Presheaf is terminal iff it maps every object to a singleton

I've read the statement in the title in a text as a side note without proof. One way is easy: using Yoneda, terminality of a presheaf $X \in [\mathcal{C}^\text{op}, \textbf{Set}]$ implies that for any ...
Jos van Nieuwman's user avatar
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Dualising the Yoneda lemma.

Let $\mathcal{C}$ be a small category, and $A, B \in \mathcal{C}_0$ objects of $\mathcal{C}$. Suppose that for every $X ∈ \mathcal{C}_0$ we have bijections $f_X : \text{Hom}_{\mathcal{C}}(A, X) \tilde\...
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Pulling Back Natural Transformations Along Functors

Suppose $\mathcal{F}: \mathscr{C} \rightarrow \mathscr{D}$, $\mathcal{G}: \mathscr{E} \rightarrow \mathscr{D}$, $\mathcal{G}': \mathscr{E} \rightarrow \mathscr{D}$ are functors such that $\mathcal{G}$ ...
Functorial Nonsense's user avatar
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Representing object for a functor mapping a category to composable morphisms

If I have a functor $D_n : \mathsf{Cat} \to \mathsf{Set}$ that maps a category into the set of all $n$-tuples of composable morphisms, $D_n(C) = A_1 \to A_2 \to A_3 \to \dots \to A_n$, what would its ...
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On the existence of a category of functors

On the book "Handbook of Categorical Algebra - Vol I" the author writes: "Again a careless argument would deduce the existence of a category whose objects are the functors from $\...
Davi Barreira's user avatar
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How to go about proving theorems about category theory? [closed]

Prove that the functor $V : Cat → Set$ that sends a category $\mathcal{C}$ to its set of arrows, is represented by the category ${0 \rightarrow 1}$ with two objects and a single non identity morphism ...
noCrayCray's user avatar
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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 ...
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$F\dashv G\dashv F\implies F,G$ 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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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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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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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:...
Maxim Nikitin's user avatar
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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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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'...
Jos van Nieuwman's user avatar
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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 ...
Sambo's user avatar
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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. ...
Stephen Harrison's user avatar
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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 $h_A(X) = \mathrm{Mor}(X, A)$ ...
Kazune Takahashi's user avatar
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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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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" ...
Daniel Donnelly's user avatar
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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 ...
Jendrik Stelzner's user avatar
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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}...
user11718766's user avatar
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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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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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