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Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

2
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2answers
62 views

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 ...
4
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2answers
40 views

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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1answer
34 views

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 $x\in X$, $\...
2
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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 $\...
2
votes
1answer
48 views

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 ...
3
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0answers
40 views

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 ...
1
vote
1answer
77 views

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 ...
2
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2answers
65 views

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 $X \in A$ ...
1
vote
2answers
32 views

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 ...
1
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0answers
38 views

$[\mathsf{I},\mathsf{C}]$ 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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0answers
40 views

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 $...
2
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1answer
60 views

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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vote
0answers
29 views

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 $\...
2
votes
1answer
40 views

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∶...
1
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1answer
30 views

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 $S$ ...
0
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2answers
43 views

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 : ...
4
votes
1answer
74 views

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 ...
0
votes
1answer
34 views

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 ...
22
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5answers
3k views

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 ...