I am self-studying and have a basic or naive question that follows from a simple observation. I have also included tags for type theory, etc because "evaluation" probably has a different meaning in these constructive approaches, which I'm also curious about.
Any set $X$ with elements $x_i$ can be characterized by a set denoted $(X)$ of pairs $(\star, x_i)$, alternatively denoted $(\star \mapsto x_i)$ (for all $x_i \in X$). Each one of these latter pairs encodes a distinct function from common domain $\{ \star \}$ to (each different element of) $X$. If we name such functions after their image, i.e. $(x_i):\star \mapsto x_i$, then the symbol "$f(x_i)$", traditionally interpreted as function evaluation, is reinterpreted as function composition, i.e. $f\circ (x_i)$ and the $\circ$ is often notationally suppressed. And roughly, $f = \cup_xf(x)$, every function decomposes to a unique set of singleton functions (better word?).
Question: Is there anything more to evaluation, or can I exactly equate evaluation with composition and forever de-clutter my mind of the term "evaluation"? Is the above perspective unusual or fruitful beyond reducing terminological clutter?
The notion of evaluation above – perhaps "extensional" is the right term? – means essentially subsetting the function at the chosen argument $x$. Perhaps I will see that a full answer to my question requires considering an intentional (?) or constructive perspective on "evaluation" (as in lambda calculus, etc). Perhaps only there will composition and evaluation legitimately have distinct meanings.
Edit: The notation $(x)$ seemed the most natural way to denote any function with singleton domain because it coheres with the conventional way to write evaluation as $f(x)$. The notation seems unambiguous because there is no context I can think of, other than "evaluation", where a single symbol, say $x$, is enclosed in brackets "()"? It does seem useful to notationally distinguish functions on singleton domain in some way: I would like the notation to be a mnemonic for whether the resulting function composite is a singleton function (i.e. a "value" like $f(x_i) = f \circ (x_i) = (\star, y_j)$) versus a non-singleton function. Perhaps there is a conventional notation for functions from singleton domain, other than $(x)$ and a conventional name for such functions, other than "singleton functions"?
Best Answer
That’s right, evaluation is not in any serious way distinct from composition with a function with singleton domain—or to remove all talk of elements entirely, a function whose domain is a terminal object. This kind of function can entirely replace the role of elements, which is at the heart of the categorical axiomatization of set theory called ETCS.