On (naturally) semifull and (semi)separable semifunctors

TL;DR

This paper extends the concept of functors to semifunctors, exploring properties like fullness and separability, and introduces notions such as natural semifullness, with theoretical characterizations and examples.

Contribution

It introduces and characterizes new properties of semifunctors, including semifullness and natural semifullness, and establishes theorems relating to their separability and semiadjunctions.

Findings

Characterization of natural semifull semifunctors.

Rafael-type theorems for (semi)separable semifunctors.

Maschke-type theorem for separable semifunctors.

Abstract

The notion of semifunctor between categories, due to S. Hayashi (1985), is defined as a functor that does not necessarily preserve identities. In this paper we study how several properties of functors, such as fullness, full faithfulness, separability, natural fullness, can be formulated for semifunctors. Since a full semifunctor is actually a functor, we are led to introduce a notion of semifullness (and then semifull faithfulness) for semifunctors. In order to show that these conditions can be derived from requirements on the hom-set components associated with a semifunctor, we look at "semisplitting properties" for seminatural transformations and we investigate the corresponding properties for morphisms whose source or target is the image of a semifunctor. We define the notion of naturally semifull semifunctor and we characterize natural semifullness for semifunctors that are part of a semiadjunction in terms of semisplitting conditions for the unit and counit attached to the semiadjunction. We study the behavior of semifunctors with respect to (semi)separability and we prove Rafael-type Theorems for (semi)separable semifunctors and a Maschke-type Theorem for separable semifunctors. We provide examples of semifunctors on which we test the properties considered so far.