Let A and B be monoidal categories and let R : A -> B be a lax monoidal functor. If R has a left adjoint L, it is well-known that the two adjoints induce functors (R) over bar = Alg(R) : Alg(A) -> Alg(B) and (L) under bar = Coalg(L) : Coalg(B) -> Coalg (A) respectively. The pair (L, R) is called liftable if the functor R has a left adjoint and if the functor (L) under bar has a right adjoint. A pleasing fact is that, when A, B and R are moreover braided, a liftable pair of functors as above gives rise to an adjunction at the level of bialgebras. In this note, sufficient conditions on the category A for (R) over bar to possess a left adjoint, are given. Natively these conditions involve the existence of suitable colimits that we interpret as objects which are simultaneously initial in four distinguished categories (among which the category of epiinduced objects), allowing for an explicit construction of (L) under bar, under the appropriate hypotheses. This is achieved by introducing a relative version of the notion of weakly coreflective subcategory, which turns out to be a useful tool to compare the initial objects in the involved categories. We apply our results to obtain an analogue of Sweedler's finite dual for the category of vector spaces graded by an abelian group G endowed with a bicharacter. When the bicharacter on G is skew-symmetric, a lifted adjunction as mentioned above is explicitly described, inducing an auto-adjunction on the category of bialgebras "colored" by G.

Liftable pairs of functors and initial objects

Ardizzoni, A;
2023-01-01

Abstract

Let A and B be monoidal categories and let R : A -> B be a lax monoidal functor. If R has a left adjoint L, it is well-known that the two adjoints induce functors (R) over bar = Alg(R) : Alg(A) -> Alg(B) and (L) under bar = Coalg(L) : Coalg(B) -> Coalg (A) respectively. The pair (L, R) is called liftable if the functor R has a left adjoint and if the functor (L) under bar has a right adjoint. A pleasing fact is that, when A, B and R are moreover braided, a liftable pair of functors as above gives rise to an adjunction at the level of bialgebras. In this note, sufficient conditions on the category A for (R) over bar to possess a left adjoint, are given. Natively these conditions involve the existence of suitable colimits that we interpret as objects which are simultaneously initial in four distinguished categories (among which the category of epiinduced objects), allowing for an explicit construction of (L) under bar, under the appropriate hypotheses. This is achieved by introducing a relative version of the notion of weakly coreflective subcategory, which turns out to be a useful tool to compare the initial objects in the involved categories. We apply our results to obtain an analogue of Sweedler's finite dual for the category of vector spaces graded by an abelian group G endowed with a bicharacter. When the bicharacter on G is skew-symmetric, a lifted adjunction as mentioned above is explicitly described, inducing an auto-adjunction on the category of bialgebras "colored" by G.
2023
72
1879
1918
https://arxiv.org/abs/1702.00224
Monoidal categories; Liftable pairs; Initial objects; Weakly coreflective subcategories; Group graded vector spaces
Ardizzoni, A; Goyvaerts, I; Menini, C
File in questo prodotto:
File Dimensione Formato  
1702.00224.pdf

Accesso aperto

Descrizione: arXiv
Tipo di file: PREPRINT (PRIMA BOZZA)
Dimensione 488.48 kB
Formato Adobe PDF
488.48 kB Adobe PDF Visualizza/Apri

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/2318/1887477
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? 1
social impact