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".

  • 1
    $\begingroup$ I don't recall any particular notation for this sort of class. $\endgroup$
    – Asaf Karagila
    Aug 30, 2017 at 11:06
  • $\begingroup$ Isn't it $Set/S$, in a categorical notation ? $\endgroup$ Aug 30, 2017 at 11:14
  • $\begingroup$ @Max What is the concept called that this notation represents in general? $\endgroup$
    – Cloudscape
    Aug 30, 2017 at 11:24
  • $\begingroup$ 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. $\endgroup$
    – drhab
    Aug 30, 2017 at 11:44
  • $\begingroup$ It's colled the slice category over $S$ $\endgroup$ Aug 30, 2017 at 13:04

1 Answer 1


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: $$\text{Ob}(\mathbf{Set}\downarrow S)$$

Of course this is only recommendable if it is presented to people who are familiar with these concepts.

  • $\begingroup$ Nice! I was previously unfamiliar with comma categories, but I quite like the concept. $\endgroup$
    – Cloudscape
    Aug 30, 2017 at 13:02

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