Academic Journal
Liftable pairs of functors and initial objects
| العنوان: | Liftable pairs of functors and initial objects |
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| المؤلفون: | Ardizzoni, Alessandro, Goyvaerts, Isar, Menini, Claudia |
| المساهمون: | Università degli Studi di Torino |
| المصدر: | Rendiconti del Circolo Matematico di Palermo Series 2 ; volume 72, issue 3, page 1879-1918 ; ISSN 0009-725X 1973-4409 |
| بيانات النشر: | Springer Science and Business Media LLC |
| سنة النشر: | 2022 |
| الوصف: | Let $$\mathcal {A}$$ A and $$\mathcal {B}$$ B be monoidal categories and let $$R:\mathcal {A}\rightarrow \mathcal {B}$$ 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 $${\overline{R}}={\textsf {Alg}}(R):\textsf {Alg}({\mathcal {A}})\rightarrow {\textsf {Alg}}({\mathcal {B}})$$ R ¯ = Alg ( R ) : Alg ( A ) → Alg ( B ) and $${\underline{L}}={\textsf {Coalg}}(L):{ \textsf {Coalg}}({\mathcal {B}})\rightarrow \textsf {Coalg}({\mathcal {A}})$$ L ̲ = Coalg ( L ) : Coalg ( B ) → Coalg ( A ) respectively. The pair ( L , R ) is called liftable if the functor $${\overline{R}}$$ R ¯ has a left adjoint and if the functor $${\underline{L}}$$ L ̲ has a right adjoint. A pleasing fact is that, when $$\mathcal {A}$$ A , $$\mathcal {B}$$ 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 $$\mathcal {A}$$ A for $${\overline{R}}$$ R ¯ 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 epi-induced objects), allowing for an explicit construction of $${\overline{L}}$$ L ¯ , 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 . |
| نوع الوثيقة: | article in journal/newspaper |
| اللغة: | English |
| DOI: | 10.1007/s12215-022-00765-4 |
| DOI: | 10.1007/s12215-022-00765-4.pdf |
| DOI: | 10.1007/s12215-022-00765-4/fulltext.html |
| الاتاحة: | http://dx.doi.org/10.1007/s12215-022-00765-4 https://link.springer.com/content/pdf/10.1007/s12215-022-00765-4.pdf https://link.springer.com/article/10.1007/s12215-022-00765-4/fulltext.html |
| Rights: | https://creativecommons.org/licenses/by/4.0 ; https://creativecommons.org/licenses/by/4.0 |
| رقم الانضمام: | edsbas.F2969269 |
| قاعدة البيانات: | BASE |
| DOI: | 10.1007/s12215-022-00765-4 |
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