The notion of semifunctor between categories, due to [9], 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 tranformations 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.

On (naturally) semifull and (semi)separable semifunctors / Bottegoni, Lucrezia. - In: JOURNAL OF ALGEBRA AND ITS APPLICATIONS. - ISSN 0219-4988. - ELETTRONICO. - (In corso di stampa). [10.1142/s0219498825502111]

On (naturally) semifull and (semi)separable semifunctors

Bottegoni, Lucrezia
In corso di stampa

Abstract

The notion of semifunctor between categories, due to [9], 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 tranformations 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.
File in questo prodotto:
File Dimensione Formato  
semifunctors-accepted.pdf

embargo fino al 20/03/2025

Tipologia: 2. Post-print / Author's Accepted Manuscript
Licenza: PUBBLICO - Tutti i diritti riservati
Dimensione 549.05 kB
Formato Adobe PDF
549.05 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
semifunctors-published-version.pdf

non disponibili

Tipologia: 2a Post-print versione editoriale / Version of Record
Licenza: Non Pubblico - Accesso privato/ristretto
Dimensione 621.31 kB
Formato Adobe PDF
621.31 kB Adobe PDF   Visualizza/Apri   Richiedi una copia
Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2992251