Previously we discussed categories with finite biproducts, or semiadditive categories. Today, partially as a further warmup for the axioms defining an abelian category, we’ll discuss monomorphisms and epimorphisms.
Monomorphisms and epimorphisms are supposed to be a categorical generalization of the familiar notion of an injective resp. surjective structure-preserving map (such as an injective resp. surjective group homomorphism or an injective resp. surjective continuous function). This idea more or less works out for monomorphisms, but epimorphisms are somewhat infamous for behaving in unexpected ways, and even monomorphisms can behave unexpectedly sometimes.
. The kind which is covariantly functorial is some variation on the group algebra ![k[G]](https://s0.wp.com/latex.php?latex=k%5BG%5D&bg=ffffff&fg=333333&s=0&c=20201002)
, which is the free 
-module on 
of functions 
with pointwise multiplication. 
and when ![\mathbb{C}[G]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BC%7D%5BG%5D&bg=ffffff&fg=333333&s=0&c=20201002)
, the corresponding state is a noncommutative refinement of 
, the corresponding state is by definition integration with respect to normalized Haar measure on 
it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on 
of (finite-dimensional continuous unitary) representations of 
being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on 
-algebra ![\mathbb{C}[x]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BC%7D%5Bx%5D&bg=ffffff&fg=333333&s=0&c=20201002)
. Along the way we will run into and connect some beautiful and classical mathematical objects.
-enriched).
):
Annoying Precision 