English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

Liftable pairs of functors and Initial objects