What structure does the space of functions into $X$ (or the cartesian exponentiation of $X$) inherit from $X$?

Question

When dealing with a space $X$, that posses a lot of structure (complete lattice, complete metric space, vector space), what can be said about the cartesian exponentiation $X^Y=\{f \mid f:Y\rightarrow X\}$ (whic is the set of functions from another set $Y$ to $X$)?

It is intuitivly clear, that the point-wise versions of operations and relations in $X$ carry over many properties of $X$ to $X^Y$. (e.g. $X^Y$ is also a vector space with point-wise addition and multiplication, it is also a complete lattice with the point-wise or product order. It is also a complete metric space under the maximum (or point-wise) metric).

Is there a field of mathematics, that is concerned with this particular sharing of structure between a space $X$ and its function space $X^Y$?

Or more pragmatically, what results can I use, in order to spare me the proof of these 16 mostly trivial and shallow implications (8 for vector space, 3 for partial order + 1 completeness, 3 for metric + 1 completemess)?

Answer 1

For algebraic structures (e.g. vector spaces, complete lattices) and relational structures (e.g. posets), or more generally models of a cartesian theory $\mathbb{T}$ (e.g. categories), the product of an arbitrary family of $\mathbb{T}$-models again a $\mathbb{T}$-model with componentwise operations and relations. This is easiest to understand in the case where $\mathbb{T}$ is an algebraic theory, i.e. a first-order theory over a signature with no relations, whose axioms are quantifier-free equations: it is clear that an equation involving componentwise operations is true if and only if it is true componentwise.

However, metric spaces are not structures of this kind. The fact that the product of finitely many metric spaces is again a metric space in a natural way is not explainable by general nonsense – and indeed, one sign that something subtle is happening is the fact that the sup metric on the product of infinitely many metric spaces does not in general induce the product topology. (At any rate, the product metric is not an example of a "componentwise" structure.)

It's also worth mentioning that the set of structure-preserving maps between two structures of the same type is in general not a structure of that type under componentwise operations. For example, although it is true that the set of homomorphisms between two abelian groups is again an abelian group, it is not true in general that the set of homomorphisms between two rings is again a ring. In fact, for one-sorted algebraic theories $\mathbb{T}$, one can show that a necessary (but not sufficient) condition for the set of $\mathbb{T}$-homomorphisms to be a $\mathbb{T}$-model is that $\mathbb{T}$ has at most one constant symbol (up to provable equality).

Answer 2

The slice category $\mathbf{C}/C$ is the category of arrows with codomain $C$ and "commutative triangles" between them. In $\mathbf{Set}$, one can think to $\mathbf{Set}/X$ as the category of $X$-valued functions.

A thoerem of Freyd (Theorem 2.1 of Toposes, Triples and Theories) states that the property of $\mathbf{C}$ being a topos (which $\mathbf{Set}$ enjoys) is inherited by $\mathbf{C}/C$. So the slice category has a lot of structure, notably finite (co-)completeness, existence of exponentials, suboject classifier as well as the Heyting lattice structure of the subobjects of the terminal object. For a richly exemplified book on topoi, see Goldblatt's book.

If your space of interest is a topos, using the previous result may translates some of the properties you are interested in.