A first-order theory is stable iff its type space is simplicially contractible

Abstract

A definable type of a first-order theory is the same as a section (retraction) of the simplicial path space (decalage) of its space of types viewed as a simplicial topological space; as is well-known, in the category of simplicial sets such sections correspond to homotopies contracting each connected component. Without the simplicial language this is stated in Exercise 8.3.3 in the model theory textbook [Tent-Ziegler], which defines a bijection between the set of all $1$-types definable over a parameter set $B$ and the set of all "coherent" families of continuous sections $\pi_n:S^T_n(B)\to S^T_{n+1}(B)$ where $S^T_n(B)$ is the Stone space of types with $n$ variables of the theory $T$ with parameters in $B$. Thus the definition of stability ``each type is definable'' says that {a first order theory is stable iff its space of types is simplicially contractible}, in the precise sense that the simplicial type space functor $\mathbb{S}^T_\bullet(B):\Delta^{op}\to {\rm Top}$, $n\longmapsto \mathbb{S}_{n+1}(B)$ fits into a certain well-known simplicial diagram in the category of simplicial topological spaces which does define contractibility for fibrant simplicial sets. In this note we spell out this and similar diagrams representing notions in model theory such as a parameter set and a type, a type being invariant, definable, and product of invariant types, and give pointers to the same diagrams in homotopy theory.