So, I'm currently reading "Category Theory for the Sciences" by David Spivak, and I'm having a little trouble understanding how congruences on monoids work.
Here's the definition, and a couple of propositions from the book:
Definition 4.1.1.17
Let $\mathcal{M} := (M, e, *)$ be a monoid. A congruence on $\mathcal{M}$ is an equivalente relation $\sim$ on $M$, such that for any $m, m' \in M$ and any $n, n' \in M$, if $m \sim m'$ and $n \sim n'$, then $m * n \sim m' \sim n'$.
Proposition 4.1.1.18 Suppose that $\mathcal{M} := (M, e, *)$ is a monoid. Then the following facts hold:
- Given any relation $R \subseteq M \times M$, there is a smallest congruence $S$ containing $R$. We call $S$ the congruence generated by $R$.
- If $R = \emptyset$ and $\sim$ is the congruence it generates, then there is an isomorphism $M \overset{\cong}{\to} (M/\sim)$.
It's the second proposition I don't get. I came up with a concrete example to try to make sense of it, to no avail:
Let $M := \mathbb{Z}$, $e := 0$, and $*:= +$ (integer addition).
For $\sim$ to be a congruence on $M$, then $m * n \sim m' \sim n'$ must hold for any $m \sim m'$ and $n \sim n'$
So, for $m = m' := 2$ and $n = n' := 3$, we have: $2 * 3 = 5 \sim 2 \sim 3$
Therefore, the congruence $\sim$ must contain the pairs: (2, 2), (3, 3), (5, 5), (2, 3), (3, 2), (2, 5), (5, 2), (3, 5), (5, 3), and the quotient $M/\sim$ must contain the equivalence class $\{2,3,5\}$. If this is so, how can there be an isomorphism $M \overset{\cong}{\to} (M/\sim)$? What did I miss?
Please bear in mind I'm still learning, so a simple explanation would be very much appreciated.
If that is the definition in the book, it is wrong. It should be:
$$m*n\sim m'*n'$$
not $m*n\sim m'\sim n'$.
Probably a typesetting error.