Question about Smiley Face notation

Question

I am reading David Spivak's "Categoty Theory For The Sciences" and in section 2.1.2.9 he uses some notation that I don't understand. He uses a Smiley Face to represent an element in a Set. Is he being cute or is it something else? Below is the text (I use a 'SF' instead of the smiley face glyph because my MathJax skills are not good).

Notation 2.1.2.9: Let X be a set and x $\in$ X an element. There is a function {SF} $\rightarrow$ X that sends SF $\mapsto$ x. We say that this function represents x element of X. We may denote it x:{SF} $\rightarrow$ X.

What does it mean to send SF $\mapsto$ X? Is he saying that the function {SF} $\rightarrow$ X sends any set with one element to X? Thanks.

Answer 1

The fact that $f$ is a function from a set $A$ to a set $B$ can be expressed as $f\colon A\to B$ (or, if we do not want to name the function, just $A\to B$). However, this does not specify how the function determines an element of $B$ for every given element of $A$. To express this, we either write something like $f(a)=b$, or we use a different arrow and write $a\mapsto b$. Of course, do completely describe $f$, we must coever every element of $A$. Fortunately, if $A=\{☺\}$ is just a singleton, specifying the result for the single element $☺$ suffices. So we could define $f$ by saying $f(☺)=x$, or equivalently (but without the need to have a name $f$ for the function) by saying $☺\mapsto x$.

Answer 2

He picked {SF} as a set of some elements (presumably smiley faces). The Exercise 2.1.2.10 with solution that follows the notation gives more clarity. It uses $f:Y\rightarrow X$ instead of $f:\{SF\}\rightarrow Y$.

Answer 3

He is defining something that I would call ”$A$-valued point“. He is considering an arbitrary set $\lbrace :) \rbrace$ of cardinality $1$. You can literally put anything inside. It does not matter (that is probably why he chose something like a smiley). Specifying an element $a \in A$ is now the same as a function $\lbrace :) \rbrace \rightarrow A$, namely by sending $:)$ to the element $a$. Furthermore, he then represents the image $f(a)$ of an element $a$ under a map $f \colon A \rightarrow B$ as $:) \mapsto a \mapsto b$, i.e. as a composition. This makes sense as you would define a map by specifying where a point $a$ is sent to and we just discussed how we can specify a point.

To define a point like this is something that one does in algebraic geometry for example. The idea is that one can understand isomorphism classes of an object $X$ by understanding all morphisms out of or into $X$ by the so-called Yoneda lemma. Therefore it can be very useful to define points as morphisms.