Why epis are surjective in $\Sigma$-Str

Question

Let $\Sigma$ be a signature. How can I see that epis are surjective and monos are injective in the concrete category $\Sigma-\mathbf{Str}$ of $\Sigma$ structures and homomoprhisms ?

Answer 1

I'll address the monic side of the equation here.

Suppose $m:\mathcal{A}\rightarrow\mathcal{B}$ is monic - that is, for every $g_1,g_2:\mathcal{X}\rightarrow\mathcal{A}$ we have $$mg_1=mg_2\implies g_1=g_2.$$ Further, towards contradiction suppose $m(a_1)=m(a_2)=b$ for some $a_1\not=a_2$.

• Phrasing this as a contradiction argument is of course unnecessary but in my opinion makes it easier to come up with initially.

What we'd love to do is take $\mathcal{X}$ to consist of a single element $x$ and look at the maps $g_1$ and $g_2$ sending $x$ to $a_1$ and $a_2$, respectively, since then we would trivially have $g_1\not=g_2$ but $mg_1=mg_2$. Of course this doesn't work (why?), but it's the idea we'll build on.

Specifically, consider the following two special cases:

• $\Sigma=\{R\}$ for some $n$-ary relation symbol $R$. In this case, we take $\mathcal{X}$ to have a single element $x$ and set $R^\mathcal{X}=\emptyset$. Then both of the maps $g_1:x\mapsto a_1$ and $g_2:x\mapsto a_2$ described above are in fact homomorphisms.

  • Remember that a homomorphism can "turn on" a relation but cannot "turn off" a relation, so it was important to set $R^\mathcal{X}=\emptyset$ rather than $R^\mathcal{X}=\{(x,...,x)\}$.

• $\Sigma=\{f\}$ for some $n$-ary function symbol $f$. In this case things are more complicated: the only one-element $\Sigma$-structure has $f(x,...,x)=x$, but it's possible for example that $f(a_1,...,a_1)\not=a_1$ in which case $x\mapsto a_1$ would not be a homomorphism. So we (probably) can't have $\mathcal{X}$ consist of a single element. Instead, we want to shift attention from individual elements to structures generated by individual elements: take $\mathcal{X}$ to be the free $\Sigma$-algebra on one generator $x$, and look at the maps induced by $x\mapsto a_1$ and $x\mapsto a_2$.

  • This isn't trivial! I've elided a lot of information in the phrase "look at." In fact, this breaks down in (say) Div, the full-and-faithful subcategory of $\{e,*,^{-1}\}$-Struc consisting of the divisible abelian groups. Towards building an intuition here, note that the inclusion $\mathbb{Z}\hookrightarrow\mathbb{Q}$ witnesses the non-monic-ness of the quotient map $q:\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}$ in $\{e,*,^{-1}\}$-Struc (or indeed Grp), but $q$ is monic in Div.

Generalizing the above to arbitrary $\Sigma$ - and whipping up the analogous argument for the epi side - is a good exercise.