@Aaron

@AaronMazel-Gee Thanks an no worries. The latter question is more general. Fix k a field. If F_A : Alg_k -> Sets is representable, and the representing object T_A is functorial in A, and nice enough for its left Kan extension to exist LT_A to exist, then even when extending to F_A : sAlg_k -> sSet is representable, is this representing object the same as LT_A? Please follow up if this still isn't phrased correctly.

I have a specific F_A in mind, but was wondering if this was true very generally. In the case F_k = Der_k(A,-), which is represented by Omega_{A/k}, the answer is yes as we discussed (so I guess I'm thinking of RF_A the derived functor).

(*Thanks and no worries.)

(also typo, F_A = Der_k(A,-))

(sorry one more clarification, left Kan extension along free algebras)

@AlexanderCampbell I hope this to be true, although for example what would be the generating (acylic) cofibrations for simplicial sets? I am expecting that the answer may possibly be linked to the arrow category of C being locally presentable

@lemiller i'm a bit confused. it sounds like you have a functor $Alg_k \xrightarrow{F_{(-)}} Fun(Alg_k,Set)$, which factors as $Alg_k \xrightarrow{T_{(-)}} Alg_k \xrightarrow{Yo} Fun(Alg_k,Set)$ [but already my variance is off!]. and for each $A$, you are contemplating taking the restriction and then left Kan extension of $F_A$ along $FreeAlg_k \to Alg_k$, and this might again be (i'd call it "co")representable by an object $LT_A$, and then.... what exactly is the question?

@ScottBalchin For any locally presentable category, the set of maps 0 --> A and A + A --> A, for A belonging to some small strongly generating set, is a generating set of cofibrations, and the empty set is a generating set of trivial cofibrations.

So for the trivial model structure on simplicial sets, the set of maps 0 --> Delta[n] and Delta[n] + Delta[n] --> Delta[n] is a generating set of cofibrations.

@AlexanderCampbell Thanks for this information, it is new to me!