This question is about Category theory for the sciences (by David Spivak).
In Exercise 3.1.2.4-a the set $T = \{x \in \mathbb{R} \; | \; 0 \leq x < 12\}$ needs to be defined using a coequalizer. I think this can be done using $Coeq(f,g)$ where $f(x) = x$ and $g(x) = x + 12$.
Then, part b is:
For any $x \in \mathbb{R}$, realize the mapping $x \cdot -: T \rightarrow T$, implied by Example 3.1.2.3, using the universal property of coequalizers.
In part c I need to show that this is an action. Example 3.1.2.3 defines a set $S$ similar to $T$ but for natural numbers, and an action is defined which for a pair $(n, s)$ gives the remainder of dividing $n + s$ by 12.
This confuses me because in the context it seems an action would be of the form $R \times T \rightarrow T$. Also, I read $\cdot -$ as "times minus", and I don't think the conditions on an action hold when we do that with the additive monoid of real numbers.
I think the notation means something different from how I read it, maybe someone can help me read it correctly?
Thanks!
If you have for example a mapping $f : A \times B \to C$ and an $a\in A$, then $f(a,-)$ is the mapping: $$f(a,-) : B \to C, b \mapsto f(a,b)$$
In your case, we have:
$$x\cdot - : T \to T, t\mapsto x\cdot t$$
which ''comes'' from an $x\in R$ and the map $\cdot : R \times T \to T$.