nLab simplicial Lawvere theory




The definition of (the syntactic category of) a Lawvere theory as a category with certain properties has an immediate generalization to simplicial categories.



Let Γ=(Skel(FinSet */))\Gamma = (Skel(FinSet^{\ast/})) be Segal's category, the opposite category of a skeleton of finite pointed sets.

A simplicial Lawvere theory is a a pointed simplicial category TT equipped with a functor i:ΓTi \;\colon\;\Gamma \to T such that

  1. TT has the same set of objects as Γ\Gamma;

  2. ii is the identity on objects

1 ii preserves finite products

Given a simplicial theory TT, then a simplicial TT-algebra is a product preserving simplicial functor XX to the simplicial category of pointed simplicial sets. The simplicial set

X(1 +)sSet X(1_+) \in sSet

(the value on the pointed 2-element set) is called the underlying simplicial set of the TT-algebra.

A homomorphism of TT-algebras is a simplicial natural transformation between such functors. Write

TAlgsSetCat T Alg \in sSet Cat

for the resulting simplicial category.

A homomorphism is called a weak equivalence or a fibration if on underlying simplicial sets it is a weak equivalence or fibration, respectively, in the classical model structure on simplicial sets. Write

(TAlg) proj (T Alg)_{proj}

for the category equipped with these classes of morphisms.

(Schwede 01, def. 2.1 and def. 2.2 and beginning of section 3)


For TT a simplicial Lawvere theory (def. ) the category (TAlg) poj(T Alg)_{poj} from def. is a simplicial model category.

This is due to (Reedy 74, theorem I), reviewed in (Schwede 01). For more see at model structure on simplicial algebras.

The analogous statement with the classical model structure on simplicial sets replaced by the classical model structure on topological spaces is due to (Schwänzl-Vogt 919


Last revised on July 22, 2021 at 08:28:34. See the history of this page for a list of all contributions to it.