nLab simplicial Lawvere theory

Idea

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

Details

Definition

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

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

1. $T$ has the same set of objects as $\Gamma$;

2. $i$ is the identity on objects

3. $i$ preserves finite products

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

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

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

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

$$T Alg \in sSet Cat$$

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

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

for the category equipped with these classes of morphisms.

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

Proposition

For $T$ a simplicial Lawvere theory (def. ) the category $(T Alg)_{poj}$ 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

Related concepts

• (infinity,1)-algebraic theory

References

• Christopher Reedy, Homology of algebraic theories, Ph.D. Thesis, University of California, San Diego, 1974

• Roland Schwänzl, Rainer Vogt, The categories of $A_\infty$- and $E_\infty$-monoids and ring spaces as closed simplicial and topological model categories, Archives of Mathematics 56 (1991) 405-411 (doi:10.1007/BF01198229)

  (the model structure on simplicial algebras)

• Stefan Schwede, Stable homotopy of algebraic theories, Topology 40 (2001) 1-41 (pdf)