Rabbit Anti-Human PDZD6 (a.a. 301-400) Polyclonal antibody

A categorical model of the multiplicative and exponential fragments of intu-itionistic linear logic (MELL), known as a linear category, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by Blute and Scott's work on categories of modules of Hopf algebras as models of linear logic, we study categories of algebras of monads (also known as Eilenberg-Moore categories) as models of MELL. We define a MELL lifting monad on a linear category as a Hopf monad - in the Bruguieres, Lack, and Virelizier sense - with a special kind of mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to the category of algebras of MELL lifting monads. We explain how groups in the category of coalgebras of the monoidal coalgebra modality induce MELL lifting monads and provide a source for such groups from enrichment over abelian groups. Along the way we also define mixed distributive laws of symmetric comonoidal monads over symmetric monoidal comonads and lifting differential category structure.

The planar cell polarity (PCP) signalling pathway is essential for embryonic development because it governs diverse cellular behaviours, and 'core PCP' proteins, such as Dishevelled and Frizzled, have been extensively characterized(1-4). By contrast, the 'PCP effector' proteins, such as Intu and Fuz, remain largely unstudied(5,6). These proteins are essential for PCP signalling, but they have never been investigated in mammals and their cell biological activities remain entirely unknown. We report here that Fuz mutant mice show neural tube defects, skeletal dysmorphologies and Hedgehog signalling defects stemming from disrupted ciliogenesis. Using bioinformatics and imaging of an in vivo mucociliary epithelium, we established a central role for Fuz in membrane trafficking, showing that Fuz is essential for trafficking of cargo to basal bodies and to the apical tips of cilia. Fuz is also essential for exocytosis in secretory cells. Finally, we identified a Rab-related small GTPase as a Fuz interaction partner that is also essential for ciliogenesis and secretion. These results are significant because they provide new insights into the mechanisms by which developmental regulatory systems such as PCP signalling interface with fundamental cellular systems such as the vesicle trafficking machinery.