H. Peter Gumm (Philipps University Marburg, Germany), Free algebra functors as coalgebraic signatures

Tue, 27 Oct 2020, 1 pm MDT

Motivating our investigation, we begin by demonstrating how all sorts of transition systems studied in Computer Science - automata (deterministic, nondeterministic, probabilistic), Kripke structures, neighbourhood systems (including topological spaces) and many more - can be naturally modeled as instances of the concept of "universal coalgebra". Each of the mentioned cases requires an appropriate Set-functor replacing what we know in universal algebra as "signature".
Various preservation properties of the signature functor F determine the structure theory of the class of all F-coalgebras. In our talk we concentrate on functors parameterized by some universal algebras, be it a complete lattice or a commutative monoid, and mainly on the functor FΣ given by constructing for a set X the free Σ-Algebra FΣ(X).
FΣ preserves preimages if and only if each Σ-term which is weakly independent at a variable position x is (strongly) independent of x, (in short: Σ implies its own 'derivative' Σ'). For every permutable variety V(Σ) the free algebra functor weakly preserves kernel pairs. For the converse we exhibit a syntactic condition stating that an equation p(x,x,y) = q(x,y,y) holds if and only both p(x,y,z) and q(x,y,z) arise as substitution instances of a common term s(x,y,z,u). We give a geometrical interpretation of this fact and use it to show that if V(Σ) is n-permutable then FΣ weakly preserves kernel pairs iff V(Σ) is Mal'cev.

[slides] [video]