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Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

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Proving naturality after determining a collection of isomorphisms.

Let $F : C \to D, \ \ G : D \to C$ be two functors we're trying to prove are quasi-inverse to each other. Suppose I've proved that $F \circ G(x) \simeq \text{id}_D(x) = x$ via the collection of ...
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1answer
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$A\times B \cong B\times A$ in a category $\mathbb{C}$

I work in a category $\mathbb{C}$, and use definitions and notation from the book 'Category Theory', by Steve Awodey. I'm learning some basic category theory from Awodey's book as part of self ...
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1answer
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Algebra operations as natural transformations

Apologies in advance if the following makes little to no sense, but here goes .. Denote $m_G : G\times G\to G$ the multiplication of a group $G$. Does it make sense to think of the map $m_G$ as some ...
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2answers
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Is a natural transformation uniquely determined by a single morphism?

Let $C$ and $D$ be categories, let $F$ and $G$ be functors from $C$ to $D$, and let $\gamma$ and $\delta$ be natural transformations from $F$ to $G$. Then my question is, if $\gamma_a=\delta_a$ for ...
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2answers
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Is the identity functor naturally isomorphic to a covariant dual functor?

It is often said that vector spaces are not naturally isomorphic to dual spaces, because the dual functor is not naturally isomorphic to the identity functor. But the latter is a rather trivial ...
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Can natural transformations be viewed as functors between images of functors?

Let $C$ and $D$ be categories, and let $F,G:C\rightarrow D$ be functors. Then a natural transformation $\tau$ from $F$ to $G$ is a family of morphisms $\{\tau_x\}_{x\in C}$ where for each $x\in X$, $\...
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2answers
63 views

Can I define a functor F and a “ΔF” of sorts, which will uniquely determine a new functor?

Let $F: \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Natural transformation between $F$ and some other functor is defined as an assignment of a morphism in $\mathcal{D}$ to each object in $\...
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1answer
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How is it obvious that $\times : C \times C \to C$ is right adjoint to the diagonal functor?

This is from "Sheaves in Geometry & Logic". $\times : C \times C \to C$ is the cartesian product of two objects. So assume that finite products exist in $C$ the above is a functor. To say ...
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0answers
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Proving adjoint equivalence from natural isomorphic unit

Let $U:\mathscr{A}\rightarrow\mathscr{B}$ and $F:\mathscr{B}\rightarrow\mathscr{A}$ be two functors. Suppose there is a natural transformation $\eta:\mathbf{1}_\mathscr{B} \rightarrow UF$ such that ...
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1answer
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Riehl's “Category Theory in Context” - Exercise 3.4.i

Let $\mathsf{I}$ be a small category, let $\mathsf{C}$ be a locally small category and let $F\colon\mathsf{I\to C}$ be a functor. Emily Riehl in her book "Category Theory in Context" defines a limit ...
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2answers
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Product in the category of functors.

Let $A$ be a category and $C= Fun(A, Set)$ (i.e. the objects are functors and morphisms are natural transformations between them). I want to know if this category has a product. For given $X \in A$ ...
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2answers
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Every natural isomorphism decomposes through another

I read a proof where the author implicitly used this: Let $F,G$ be endofuctors of a category $\mathcal C$. If $\mu:F\Rightarrow G$ is a natural isomorphism, then the components of any other natural ...
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0answers
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$[\mathsf{I},\mathsf{C}]$ and $[\mathsf{J},\mathsf{C}]$ are equivalent if $\mathsf{I}$ and $\mathsf{J}$ are. [duplicate]

Let $\mathsf{I}$ and $\mathsf{J}$ be equivalent categories. Let $\mathsf{C}$ be another category. I need to prove that the categories of functors $[\mathsf{I},\mathsf{C}]$ and $[\mathsf{J},\mathsf{C}]$...
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Sandwich natural transformation between two functors

In the Kleisli adjunction we have: $G\varepsilon F = \mu$ where $\varepsilon$ is a natural transformation called the counit. How exactly is $G\varepsilon F$ defined? I understand $G\varepsilon$ and $...
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1answer
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If the Yoneda lemma essentially says that $\text{Hom}(\text{Hom}(\cdot, x), A) \simeq A(x)$, then what about higher iterates of $\text{Hom}$?

Assume that the $C$ in $\text{Hom}_C(x,y)$ can always be inferred from $x,y$ so that we can change our notation to $\text{H}(x,y) := \text{Hom}_C(x,y)$ Then the Yoneda lemma "looks at a single step ...
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0answers
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How to arrive at unique factorization through the limit given naturality compatibility conditions?

If $\alpha : I \to C$ from a small category to any category $C$. Define a functor $\lim\limits_{\rightarrow} \alpha : X \mapsto \lim\limits_{\leftarrow} \text{Hom}_C(\alpha, X)$ from $C^{op}$ to $\...
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1answer
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Proving equivalent categories have binary products if one of them does

Suppose $\mathscr{C},\mathscr{D}$ are equivalent categories. Then there exist functors $S∶ \mathscr{C}\rightarrow \mathscr{D}$ and $T∶\mathscr{D}→\mathscr{C}$ with the compositions defined $T\circ S∶...
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1answer
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Showing when two functors are naturally isomorphic, if one is faithful, then the other also is. [duplicate]

Supposing we have a natural isomorphism $\tau : S \rightarrow T$ between functors $S,T : \mathscr{C} \rightarrow \mathscr{D}$, how exactly do we show that if $S$ is faithful, then so is $T$? If $S$ ...
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Proving for two naturally isomorphic functors, if one is full, then so is the other. [duplicate]

So we let $S,T : \mathscr{C} \rightarrow \mathscr{D}$ be naturally isomorphic functors. We seek to show that if $S$ is a full functor, then so is $T$. As given, we have a natural isomorphism $\tau : ...
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understanding natural transformations that are not natural isomorphisms

What's the right way to think about what a natural transformation that is not a natural isomorphism is? How strong of a claim is it making about the relationship between the two functors it's related ...
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Showing the Naturality of Two (Isomorphic) Left Adjoints

From pg. 85 of Categories for the Working Mathematician: Problem: I understand everything except the red underlined portion. To introduce some notation, let $h: x \rightarrow x'$ in $X$. Then ...
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Uniqueness of adjoint functors up to isomorphism

Suppose we are given functors $F:\mathcal{C}\rightarrow\mathcal{D}$ and $G,G':\mathcal{D}\rightarrow\mathcal{C}$ such that $G$ and $G'$ are both right adjoint to $F$. To show that $G$ and $G'$ are ...