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Rigidification of algebras over multi-sorted theories

• Julia E. Bergner?, Rigidification of algebras over multi-sorted theories, Algebraic & Geometric Topology 6, pages 1925-1955, 2006. arXiv:0508152

Definitions

Definition. Given a set $S$, an $S$-sorted algebraic theory $\mathcal{T}$ is a small category with objects $T_{\underline{\alpha}^n}$ where $\underline{\alpha^n} = \langle \alpha_1, \dots, \alpha_n \rangle$ for $\alpha_i \in S$ and $n \geq 0$ varying, and such that each $T_{\underline{\alpha^n}}$ is equipped with an isomorphism $T_{\underline{\alpha^n}} \cong \prod_{i=1}^n T_{\alpha_i}$.

Definition. Given an $S$–sorted theory $\mathcal{T}$, a (strict simplicial) $\mathcal{T}$-algebra is a product-preserving functor $A \colon \mathcal{T} \to \mathsf{SSets}$. Here, product-preserving means that the canonical map $A(T_{\underline{\alpha}^n} ) \to \prod^n_{i=1} A(T_{\alpha_i})$, induced by the projections $T_{\underline{\alpha}^n} \to T_{\alpha_i}$ for all $1 \leq i \leq n$, is an isomorphism of simplicial sets.

Definition. Given an $S$–sorted theory $\mathcal{T}$, a homotopy $\mathcal{T}$-algebra is a functor $X \colon \mathcal{T} \to \mathsf{SSets}$ which preserves products up to homotopy, i.e. for all $\alpha \in S^n$ the canonical map $X(T_{\underline{\alpha}^n} ) \to \prod^n_{i=1} X(T_{\alpha_i})$, induced by the projections $T_{\underline{\alpha}^n} \to T_{\alpha_i}$ for all $1 \leq i \leq n$, is a weak equivalence of simplicial sets.

Main Result

The main result is a multi-sorted generalization of a theorem by Badzioch:

Theorem. Let $\mathcal{T}$ be an algebraic theory. Any homotopy $\mathcal{T}$-algebra is weakly equivalent as a homotopy $\mathcal{T}$-algebra to a strict $\mathcal{T}$-algebra.

The main result is stated:

Theorem. Let $\mathcal{T}$ be a multi-sorted algebraic theory. Any homotopy $\mathcal{T}$-algebra is weakly equivalent as a homotopy $\mathcal{T}$-algebra to a strict $\mathcal{T}$-algebra.

Examples

Several examples of multi-sorted theories are given.

• (Example 3.2) Pairs $(G,X)$ where $G$ is a group and $X$ is a set.

• (Example 3.2) Pairs $(G,X)$ as above, and an action of $G$ on $X$.

• (Example 3.3) Ring-module pairs.

• (Example 3.4) The theory for operads has a sort for each natural number, corresponding to the arity of the operation.

• (Example 3.5) Categories with a fixed object set.

References

• B. Badzioch?, Algebraic theories in homotopy theory?, Ann. of Math. (2) 155, pages 895-913, 2002.

• William Lawvere?, Functorial Semantics of Algebraic Theories? , Ph.D. thesis Columbia University (1963). Published with an author’s comment and a supplement in: Reprints in Theory and Applications of Categories 5 (2004) pp 1–121. (abstract)