# How do we denote the class of all functions with a given target?

Suppose that $S$ is a set. What is a standard notation for the class $$\{f : X \to Y | Y = S, X \text{ set}\} ?$$

Alternatively, how do we denote the class of all arbitrarily large tuples with elements in $S$?

EDIT: I would prefer a notation that does not use category theory, and I'd also be happy with "educated inventions".

• I don't recall any particular notation for this sort of class. Aug 30, 2017 at 11:06
• Isn't it $Set/S$, in a categorical notation ? Aug 30, 2017 at 11:14
• @Max What is the concept called that this notation represents in general? Aug 30, 2017 at 11:24
• Concerning your edit: then I fear that you must come up with some notation created by yourself. Let's wait and see, though. I am curious. Aug 30, 2017 at 11:44
• It's colled the slice category over $S$ Aug 30, 2017 at 13:04

If $S$ is a set then it can be interpreted as an object in category $\mathbf{Set}$.
Object $S$ induces the slice category $(\mathbf{Set}\downarrow S)$ and objects of this category are arrows in $\mathbf{Set}$ (which are functions) that have $S$ as codomain.
So you could denote the class by: $$\text{Ob}(\mathbf{Set}\downarrow S)$$