On abstract and concrete minions

Abstract

This thesis is expository in nature. We analyze the connection between abstract minions, which can be described as functors from the category of finite ordinals to sets, and concrete minions, which are sets $\mathrm{Pol}(A, B)$ of polymorphisms $A^k \to B$ between relational structures $A$ and $B$. The functorial structure arises because a function $\alpha\colon n \to k$ transforms an $n$-ary polymorphism $f\colon A^n \to B$ to the polymorphism $(x_0, \ldots, x_{k-1}) \mapsto f(x_{\alpha (0)}, \ldots, x_{\alpha (n-1)})$. This data is relevant to the constraint satisfaction problems over finite domain, because minion homomorphisms $\mathrm{Pol}(A, B) \to \mathrm{Pol}(A', B')$ give rise to log-space reductions from the promise constraint satisfaction problem $\mathrm{PCSP}(A', B')$ to $\mathrm{PCSP}(A, B)$. Thus, they are valuable to understand the homomorphism order of polymorphism minions over finite-domain structures, especially since it is unknown whether the P vs. NP complexity dichotomy of constraint satisfaction problems extends to the more generalized promise setting. Crucially, although implicitly used in some papers to solve nontrivial problems, the concept of an abstract minion has not been exposed on so far, even though much is known about functor categories. The aim of this thesis is to start closing this gap, and to apply well-known constructions from category theory to minions. Furthermore, we identify a condition under which an abstract minion can arise as a concrete one over a finite domain, and deduce that many constructions remain stable under some of these concreteness assumptions. We will conclude that some of the homomorphism orders in question are uncountable distributive bounded lattices, and moreover (bi-) Heyting algebras. Along the way, we collect open questions related to abstract and concrete minions.