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Let $X$ and $Y$ denote sets. Then if $+$ is a binary operation on $Y$, then we can obtain a new binary operation $+'$ on $Y^X$ in a canonical way as follows.

$$(f+' g)(x) = f(x)+g(x)$$

Question. The other day, I noticed a suggestive-looking variant of the above definition. However, I'm not sure what to make of this observation. Does it have any particular significance?

The variant.

Let us firstly assign to each $x \in X$ an "evaluation" function $\tilde{x} : Y^X \rightarrow Y$ with defining property $\tilde{x}(f) = f(x).$ This allows $+'$ to be defined as follows, where it is understood that $f$ and $g$ range over all functions $X \rightarrow Y$ and $x$ ranges over every element of $X$.

$$\tilde{x}(f+' g) = \tilde{x}(f)+\tilde{x}(g)$$

In other words, we're defining that $+'$ is the unique binary operation such that for all $x \in X,$ we have that $\tilde{x}$ is a magma homomorphism with source $(Y^X, +')$ and target $(Y,+)$.

Does this final characterization of $+'$ have any particular significance and/or does it "go anywhere"?

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1 Answer 1

In fact, this is part of a larger story, initiated by Freyd in his paper "Algebra valued functors in general and tensor products in particular".

If $\mathcal{A}$ is a category of algebraic structures with forgetful functor $U : \mathcal{A} \to \mathsf{Set}$, then one can define $\mathcal{A}$-objects in an arbitrary category $\mathcal{C}$ as follows: These are objects $X \in \mathcal{C}$ equipped with a factorization of $\hom(-,X) : \mathcal{C}^{op} \to \mathsf{Set}$ over $U$. In other words, $\hom(Y,X)$ acquires the structure of an object of $\mathcal{A}$, naturally in $Y$. If $\mathcal{C}$ has products, these objects can also be described internally to $\mathcal{C}$ - this is an application of the Yoneda Lemma. For example, a group object is an object $X$ equipped with morphisms $X \times X \to X$ (multiplication), $1 \to X$ (unit), $X \to X$ (inversion) such certain diagrams commute, which correspond to the usual group axioms. This is a very important notion, it includes groups, topological groups, Lie groups, and under a suitably generalization even Hopf algebras.

More simply, a magma object is an object $X$ equipped with a morphism $m : X \times X \to X$. If $Y$ is an arbitrary object, then $\hom(Y,X)$ becomes a magma, as already mentioned above. The operation is just $$\hom(Y,X) \times \hom(Y,X) \cong \hom(Y,X \times X) \xrightarrow{m_*} \hom(Y,X).$$ This is what you have observed for the category of sets.

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