# Questions tagged [natural-transformations]

Questions concerning morphisms of functors or unnatural isomorphisms.

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### Monoidal adjunction whose right-adjoint functor has structure morphisms which are epimorphisms

Let $(\mathbf{C},\otimes,1)$ and $(\mathbf{D},*,e)$ be monoidal categories and let $L:\mathbf{C}\rightarrow \mathbf{D}$ and $R:\mathbf{D}\rightarrow \mathbf{C}$ be functors. Suppose that there exists ...
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### Natural homomorphism involved in the Kunneth-Formula

I have been reading the proof of the Kunneth-Formula in MacLane's Homology book, and to prove that is natural he does this, $(10.6)$ (sorry for posting a picture but I dont know how to diagrams ...
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### Generalized diagrams

Certain mathematical objects can be described as a functor $F : \mathcal{J} \to \mathcal{C}$ from a small index category $\mathcal{J}$ to a bigger category $\mathcal{C}$. For example, we can think of ...
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### Natural isomorphism in linearly distributive categories with left and right dualities

Let $\mathrm{C}$ be a linearly distributive category with a left and right duality, i.e. it is a monodical category "twice": once for the bifunctor $\otimes:\mathrm{C}\times\mathrm{C}\to\mathrm{C}$ ...
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### Limits and colimits chasing: why $h$ is iso? [duplicate]

I do not follow why $h$ in the second snippet below is iso from the fact that $$p=q.$$ Why is it injective and why surjective ?
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### Natural transformation between $\mathcal{F}$ and $\mathcal{G}$

Consider the following functors from the category of finite Abelian groups $F AbGrps$ to the category of sets defined as follows: $\mathcal{F}: F AbGrps \to Sets$, a group $G$ is mapped to the ...
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### Self-Natural Maps for Forgetful Functor

Let $\mathscr{F}$ denote the forgetful functor from the category of groups to the category of sets. Why is there more then one natural map from $\mathscr{F}$ to $\mathscr{F}$? What are all of the ...
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### Kan Extensions and Coends

I am lost in a bit of a haze of abstraction and am wondering if someone can set me straight. I'm reading the chapter in Categories for the Working Mathematician, and Mac Lane points out on page 238 ...
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### Composition of a natural transformations with a functor--whiskering [duplicate]

If $M$ is an endofunctor on a category $\cal K$ and $\eta:Id_{\cal K}\to M$ is a natural transorfmation, what is the difference between $\eta M$ and $M\eta$, and how these two (componentwise) are ...
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### Is $H_n(X,A)$ naturally isomorphic to $\tilde{H}_n(X/A)$?

It is well known that under some assumptions on a pair $(X,A)$ of a topological space and a subspace we have $H_n(X,A)\simeq \tilde{H}_n(X/A)$. Such an assumption can be for example that $A$ is closed ...
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### $\Theta$ is induced by the inclusion map $Z_p\otimes Z_q' \to (C\otimes C')_{p+q}$?

Let $C$ and $C'$ be two chain complexes and $C\otimes C'$ their tensor product. Then define the map \begin{align} \Theta : H_p(C)\otimes H_q(C')&\to H_{p+q}(C\otimes C')\\ {[z_p]}\otimes {[z_q']}&...
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### Splitting induces isomorphisms

Let $M^n$ be a closed topological manifold. For $1 \leq q \leq n$, we have the following short exact sequence:  0 \longrightarrow \operatorname{Ext}(H_{q-1}(M),\mathbb{Z}) \stackrel{\beta}{\...
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### Natural transformation between functors $\mathbf {CRing}\to\mathbf{Mon}$

Regarding Example 1.3.5: 1) How to see that a ring homomorphism $R\to S$ induces a monoid homomorphism $M_n(R)\to M_n(S)$? It definitely induces a map, but why is it a monoid homomorphism? 2) Why do ...
My understanding from model theory is that, given groups A and B, the statement $A \cong B$ implies that any for any first order statement $P$ in the language of groups, $P(A) \iff P(B)$. Can an ...