Let P be a (topological, for example) operad, and let P be the associated bicolored operad whose algebras are triplets (A,B,f) where A and B are P-algebras and f is a morphism of such algebras. It has two colors a and b and is given by:
P(x1,…,xn;a)={P(n)∅xi=a∀i,otherwise.
P(x1,…,xn;b)=P(n)
Then if P is formal, so is P. This is to be contrasted with the fact that the Swiss-Cheese operad is not formal, even though the operad of little 2-disks is. The operad governing morphisms is formal, but not the one governing actions.
Let P be a dg-operad (maybe with some finiteness assumptions), and A be an algebra over this operad. Then the symmetric collection {A⊗n}n≥0 is a right-comodule over P∨. The coaction morphism
A⊗(k+l−1)∘i∗A⊗k⊗P∨(l)
is dual to the map P(l)⊗A⊗(k+l−1)→A⊗k given by
ρ⊗a1⊗⋯⊗ak+l−1↦a1⊗…ρ(ai,…,ai+l−1)⋯⊗ak+l−1.
If F:C×D→E is a bifunctor that preserves reflexive coequalizers in each variable, then it preserves reflexive coequalizers in the following sense: if M1d0,d1M0d0M is a reflexive coequalizer in C (with reflector s0), and M1′d0′,d1′M0′d0M′ is a reflexive coequalizer in D (with reflector s0′), then so is
F(M1,M1′)→F(M0,M0′)→F(M,M′).
This is a classical result, and the proof hinges on the following trick (that I saw in Goerss and Hopkins’ André–Quillen (co)-homology for simplicial algebras over simplicial operads, but it can probably be found elsewhere): if f equalizes F(d0,d0′) and F(d1,d1′), then it must also equalize F(1,d0′) and F(1,d1′), and it must also equalize F(d0,1) and F(d1,1). Indeed:
This then implies that if P is an operad, then the free P-algebra functor preserves reflexive coequalizers. It didn’t really click in before for me that this was the reason why.
The retract argument: if f=(XiApY) is a composite and if f has the left lifting property against p, then f is a retract of i. Similarly, if it has the right lifting property against i, then it is a retract of p. This seems to be useful for weak factorization systems, for example if f is an acyclic cofibration and you manage to factorize it as something followed by a fibration, then f is a retract of that something.