Notebook @ Functorial
Rigidification of algebras over multisorted theories (Moeller)

  • 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 SS, an SS-sorted algebraic theory 𝒯\mathcal{T} is a small category with objects T α̲ nT_{\underline{\alpha}^n} where α n̲=α 1,,α n\underline{\alpha^n} = \langle \alpha_1, \dots, \alpha_n \rangle for α iS\alpha_i \in S and n0n \geq 0 varying, and such that each T α n̲T_{\underline{\alpha^n}} is equipped with an isomorphism T α n̲ i=1 nT α iT_{\underline{\alpha^n}} \cong \prod_{i=1}^n T_{\alpha_i}.

Definition. Given an SS–sorted theory 𝒯\mathcal{T}, a (strict simplicial) 𝒯\mathcal{T}-algebra is a product-preserving functor A:𝒯SSetsA \colon \mathcal{T} \to \mathsf{SSets}. Here, product-preserving means that the canonical map A(T α̲ n) i=1 nA(T α i)A(T_{\underline{\alpha}^n} ) \to \prod^n_{i=1} A(T_{\alpha_i}), induced by the projections T α̲ nT α iT_{\underline{\alpha}^n} \to T_{\alpha_i} for all 1in1 \leq i \leq n, is an isomorphism of simplicial sets.

Definition. Given an SS–sorted theory 𝒯\mathcal{T}, a homotopy 𝒯\mathcal{T}-algebra is a functor X:𝒯SSetsX \colon \mathcal{T} \to \mathsf{SSets} which preserves products up to homotopy, i.e. for all αS n\alpha \in S^n the canonical map X(T α̲ n) i=1 nX(T α i)X(T_{\underline{\alpha}^n} ) \to \prod^n_{i=1} X(T_{\alpha_i}), induced by the projections T α̲ nT α iT_{\underline{\alpha}^n} \to T_{\alpha_i} for all 1in1 \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 𝒯\mathcal{T} be an algebraic theory. Any homotopy 𝒯\mathcal{T}-algebra is weakly equivalent as a homotopy 𝒯\mathcal{T}-algebra to a strict 𝒯\mathcal{T}-algebra.

The main result is stated:

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

Examples

Several examples of multi-sorted theories are given.

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

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

  • (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)

category: notes-moeller