Here's a breakdown of the differences between each PDF style:
| Style | Class | Font | Theorem Environments |
|---|---|---|---|
| Style 1 | book |
Alegreya Sans | tcbthm |
| Style 2 | book |
Alegreya Sans | amsthm |
| Style 3 | book |
Arno* | amsthm |
| Style 4 | book |
Computer Modern | amsthm |
*To be replaced with Linus Romer's Elemaints when it is released.
The equaliser of $\webleft (f,g\webright )$ is the pair consisting of:
given by the canonical inclusion $\text{eq}\webleft (f,g\webright )\hookrightarrow \text{Eq}\webleft (f,g\webright )\hookrightarrow X$.
We claim that $\webleft (\text{Eq}\webleft (f,g\webright ),x_{0}\webright )$ is the categorical equaliser of $f$ and $g$ in $\mathsf{Sets}_{*}$. First we need to check that the relevant equaliser diagram commutes, i.e. that we have
which indeed holds by the definition of the set $\text{Eq}\webleft (f,g\webright )$. Next, we prove that $\text{Eq}\webleft (f,g\webright )$ satisfies the universal property of the equaliser. Suppose we have a diagram of the form
in $\mathsf{Sets}_{*}$. Then there exists a unique morphism of pointed sets
making the diagram
commute, being uniquely determined by the condition
via
for each $x\in E$, where we note that $e\webleft (x\webright )\in A$ indeed lies in $\text{Eq}\webleft (f,g\webright )$ by the condition
which gives
for each $x\in E$, so that $e\webleft (x\webright )\in \text{Eq}\webleft (f,g\webright )$. Lastly, we note that $\phi $ is indeed a morphism of pointed sets, as we have
where we have used that $e$ is a morphism of pointed sets.
