Katětov Functors

Abstract

We develop a theory of Katětov functors which provide a uniform way of constructing Fraïssé limits. Among applications, we present short proofs and improvements of several recent results on the structure of the group of automorphisms and the semigroup of endomorphisms of some Fraïssé limits.

Retracts of Ultrahomogeneous Structures in the Context of Katetov Functors

“…Another characterization of retracts of Fraïssé limits in terms of categorical properties of those objects was presented in [7]. In this paper, however, we generalize the main result of [4] and provide the characterization of retracts of a large class of Fraïssé limits using the tools developed in [8], which then enable us to conclude that in many cases a structure is a retract of a Fraïssé limit if and only if it is algebraically closed in the surrounding category. This is the content of Section 3.…”

“…Moreover, the canonical embeddings η ω A : A Ñ K ω pAq constitute a natural transformation η ω : ID Ñ K ω . Thus, we have that K ω : C Ñ C is a Katětov functor as well [8].…”

“…The elegance of Katětov's construction caught the eye of the scientific community and started several new lines of research. One such spin-off is presented in [8], where we apply the idea of the Katětov's construction of the Urysohn space to a wide range of Fraïssé limits, showing thus that the Katětov's construction draws its strength from its strong categorical properties.…”

“…It was first observed in [1] that the construction K is actually functorial with respect to embeddings. However, more is true: if A is the category of all finite metric spaces with rational distances and nonexpansive maps, and C is the category of all countable metric spaces with rational distances and nonexpansive maps, then K can be turned into a functor from A to C straightforwardly [8]. This observation is then expanded to a general setting, where A is a category of all "finitely generated structures" and C is the category of colimits of ω-chains of objects from A and K is a functor A Ñ C with certain properties.…”

“…This observation is then expanded to a general setting, where A is a category of all "finitely generated structures" and C is the category of colimits of ω-chains of objects from A and K is a functor A Ñ C with certain properties. Our main result in [8] is that the existence of such a functor K : A Ñ C, which we refer to as the Katětov functor, implies that A is an amalgamation class, and that its Fraïssé limit can be constructed in the fashion of the Katětov's construction. Details are outlined in Section 2.…”