  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.
