$(LLP(Epi), Epi)$ is a WFS on any variety of algebras

Question

This entry in the Joyal catlab claims without proof that in a category $\bf V$ which is a "variety of algebras" the two classes $(LLP(Epi), Epi)$ form a weak factorization system. I interpret this claim in the following way:

1. $\bf V$ is monadic over $\bf Set$, i.e. there exists an adjunction $F\colon \mathbf{Set}\leftrightarrows \mathbf{V}\colon U$ such that $U$ is faithful and algebras for the monad $UF$ are precisely the category $\bf V$.

2. If I define $Epi\subset\hom(\mathbf V)$ to be the class of arrows $f$ such that $Uf$ is a surjective function in $\bf Set$, and $LLP(Epi)$ to be the left orthogonal of this class, then these classes are orthogonal and every arrow in $\bf V$ can be factored as a composition $X\xrightarrow{LLP(Epi)}A\xrightarrow{Epi}Y$.

The two classes are orthogonal almost by definition; it's rather easy to prove the characterization given for $LLP(Epi)$: codomain retracts of an inclusion $A\hookrightarrow A\coprod FX$, where $X$ is any set.

My problem is that I've no clue on how to prove the factorization property: the only thing I can think to appeal is some general, indirect result on existence, but I'm looking for something more explicit.

Answer 1

There are a few ways to go about this.

First, let us observe that the surjections in $\mathbf{Set}$ are precisely the maps that have the left lifting property with respect to the inclusion $\emptyset \hookrightarrow 1$. Thus, by the usual adjointness argument, the surjections in $\mathcal{V}$ are precisely the homomorphisms that have the right lifting property with respect to the induced homomorphism $F \emptyset \to F 1$. So this weak factorisation system, if it exists, must be cofibrantly generated.

If we assume $\mathcal{V}$ is an algebraic variety in the traditional sense, then $\mathcal{V}$ will also be locally finitely presentable (or at least, locally presentable). We can then run Quillen's or Garner's small object argument to get the required factorisation.

But in fact it suffices for $\mathcal{V}$ to have finite coproducts (which is automatic if you assume the axiom of choice). Then we can proceed by hand: the factorisation of a homomorphism $f : A \to B$ is $A \to A + F U B \to B$, where $A \to A + F U B$ is the coproduct insertion and $A + F U B \to B$ is defined on $A$ by $f$ and on $F U B$ by the counit. Incidentally, I think this is what Garner's small object argument produces (cf Example 3.11 in [Understanding the small object argument]).