# 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 ...
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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'...
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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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### 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. ...
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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 $h_A(X) = \mathrm{Mor}(X, A)$ ...
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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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### 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 ...
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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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### 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 ...
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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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### 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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### 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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### 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 $C_1, C_2 \in Ob(\mathcal{C})$, $F(C_1) = F(C_2) \implies G(C_1) = G(C_2)$, ...
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### 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 ...
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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
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### 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. ...
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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 ...
### 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(\...