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Michael Hardy
Michael Hardy
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We consider the category of endofunctors of finite sets with natural transformations.

Is there an infinite chain $F_1$, $F_2$, ...$F_1, F_2, \ldots$ of endofunctors such that there is a natural transformation from $F_i$ to $F_{i+1}$ for all $i\in \mathbb{N}$ but no natural transformation from $F_{i+1}$ to $F_i$ for any $i\in \mathbb{N}$?

(This problem has a relevance in universal algebra as one can consider for each pair of finite structures $A, B$ the functor mapping the set $\{ 1,2,...,n\}$ to the set of homomorphisms $\operatorname{Hom}(A^n,B)$ with the obvious maps on morphisms. In fact, this was the motivation for the question.)

We consider the category of endofunctors of finite sets with natural transformations.

Is there an infinite chain $F_1$, $F_2$, ... of endofunctors such that there is a natural transformation from $F_i$ to $F_{i+1}$ for all $i\in \mathbb{N}$ but no natural transformation from $F_{i+1}$ to $F_i$ for any $i\in \mathbb{N}$?

(This problem has a relevance in universal algebra as one can consider for each pair of finite structures $A, B$ the functor mapping the set $\{ 1,2,...,n\}$ to the set of homomorphisms $\operatorname{Hom}(A^n,B)$ with the obvious maps on morphisms. In fact, this was the motivation for the question.)

We consider the category of endofunctors of finite sets with natural transformations.

Is there an infinite chain $F_1, F_2, \ldots$ of endofunctors such that there is a natural transformation from $F_i$ to $F_{i+1}$ for all $i\in \mathbb{N}$ but no natural transformation from $F_{i+1}$ to $F_i$ for any $i\in \mathbb{N}$?

(This problem has a relevance in universal algebra as one can consider for each pair of finite structures $A, B$ the functor mapping the set $\{ 1,2,...,n\}$ to the set of homomorphisms $\operatorname{Hom}(A^n,B)$ with the obvious maps on morphisms. In fact, this was the motivation for the question.)

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YCor
YCor
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Is there an infinite chain of Endofunctorsendofunctors of Finitefinite sets?

We consider the category of endofunctors of finite sets with natural transformations.

Is there an infinite chain $F_1$, $F_2$, ... of endofunctors such that there is a natural transformation from $F_i$ to $F_{i+1}$ for all $i\in \mathbb{N}$ but no natural transformation from $F_{i+1}$ to $F_i$ for any $i\in \mathbb{N}$?

(This problem has a relevance in universal algebra as one can consider for each pair of finite structures $A, B$ the functor mapping the set $\\\{ 1,2,...,n\\\}$$\{ 1,2,...,n\}$ to the set of Homomorphismshomomorphisms $\operatorname{Hom}(A^n,B)$ with the obvious maps on morphisms. In fact, this was the motivation for the question.)

Is there an infinite chain of Endofunctors of Finite sets?

We consider the category of endofunctors of finite sets with natural transformations.

Is there an infinite chain $F_1$, $F_2$, ... of endofunctors such that there is a natural transformation from $F_i$ to $F_{i+1}$ for all $i\in \mathbb{N}$ but no natural transformation from $F_{i+1}$ to $F_i$ for any $i\in \mathbb{N}$?

(This problem has a relevance in universal algebra as one can consider for each pair of finite structures $A, B$ the functor mapping the set $\\\{ 1,2,...,n\\\}$ to the set of Homomorphisms $\operatorname{Hom}(A^n,B)$ with the obvious maps on morphisms. In fact, this was the motivation for the question.)

Is there an infinite chain of endofunctors of finite sets?

We consider the category of endofunctors of finite sets with natural transformations.

Is there an infinite chain $F_1$, $F_2$, ... of endofunctors such that there is a natural transformation from $F_i$ to $F_{i+1}$ for all $i\in \mathbb{N}$ but no natural transformation from $F_{i+1}$ to $F_i$ for any $i\in \mathbb{N}$?

(This problem has a relevance in universal algebra as one can consider for each pair of finite structures $A, B$ the functor mapping the set $\{ 1,2,...,n\}$ to the set of homomorphisms $\operatorname{Hom}(A^n,B)$ with the obvious maps on morphisms. In fact, this was the motivation for the question.)

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Sebastian Meyer
Sebastian Meyer
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Is there an infinite chain of Endofunctors of Finite sets?

We consider the category of endofunctors of finite sets with natural transformations.

Is there an infinite chain $F_1$, $F_2$, ... of endofunctors such that there is a natural transformation from $F_i$ to $F_{i+1}$ for all $i\in \mathbb{N}$ but no natural transformation from $F_{i+1}$ to $F_i$ for any $i\in \mathbb{N}$?

(This problem has a relevance in universal algebra as one can consider for each pair of finite structures $A, B$ the functor mapping the set $\\\{ 1,2,...,n\\\}$ to the set of Homomorphisms $\operatorname{Hom}(A^n,B)$ with the obvious maps on morphisms. In fact, this was the motivation for the question.)