Kan extensions and the calculus of modules for ∞\infty-categories

Various models of $(\infty,1)$-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an $\infty$-cosmos. In a generic $\infty$-cosmos, whose objects we call $\infty$-categories, we introduce modules (also called profunctors or correspondences) between $\infty$-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from $A$ to $B$ is an $\infty$-category equipped with a left action of $A$ and a right action of $B$, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed $\infty$-cosmoi, to limits and colimits of diagrams valued in an $\infty$-category, as introduced in previous work.Comment: 84 pages; a sequel to arXiv:1506.05500; v2. new results added, axiom circularity removed; v3. final journal version to appear in Alg. Geom. To

Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions

Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasi-category admitting these (co)limits and that they are preserved by the functors in the limit cone. In particular, the Bousfield-Kan homotopy limit of a diagram of quasi-categories admit any limits or colimits existing in and preserved by the functors in that diagram. In previous work, we demonstrated that the quasi-category of algebras for a homotopy coherent monad could be described as a weighted limit with projective cofibrant weight, so these results immediately provide us with important (co)completeness results for quasi-categories of algebras. These generalise most of the classical categorical results, except for a well known theorem which shows that limits lift to the category of algebras for any monad, regardless of whether its functor part preserves those limits. The second half of this paper establishes this more general result in the quasi-categorical setting: showing that the monadic forgetful functor of the quasi-category of algebras for a homotopy coherent monad creates all limits that exist in the base quasi-category, without further assumption on the monad. This proof relies upon a more delicate and explicit analysis of the particular weight used to define quasi-categories of algebras.Comment: 33 pages; a sequel to arXiv:1306.5144 and arXiv:1310.8279; v3: final journal version with updated internal references to the new version of "Homotopy coherent adjunctions and the formal theory of monads

DNAAF1 links heart laterality with the AAA+ ATPase RUVBL1 and ciliary intraflagellar transport

DNAAF1 (LRRC50) is a cytoplasmic protein required for dynein heavy chain assembly and cilia motility, and DNAAF1 mutations cause primary ciliary dyskinesia (PCD; MIM 613193). We describe four families with DNAAF1 mutations and complex congenital heart disease (CHD). In three families, all affected individuals have typical PCD phenotypes. However, an additional family demonstrates isolated CHD (heterotaxy) in two affected siblings, but no clinical evidence of PCD. We identified a homozygous DNAAF1 missense mutation, p.Leu191Phe, as causative for heterotaxy in this family. Genetic complementation in dnaaf1-null zebrafish embryos demonstrated the rescue of normal heart looping with wild-type human DNAAF1, but not the p.Leu191Phe variant, supporting the conserved pathogenicity of this DNAAF1 missense mutation. This observation points to a phenotypic continuum between CHD and PCD, providing new insights into the pathogenesis of isolated CHD. In further investigations of the function of DNAAF1 in dynein arm assembly, we identified interactions with members of a putative dynein arm assembly complex. These include the ciliary intraflagellar transport protein IFT88 and the AAA+ (ATPases Associated with various cellular Activities) family proteins RUVBL1 (Pontin) and RUVBL2 (Reptin). Co-localization studies support these findings, with the loss of RUVBL1 perturbing the co-localization of DNAAF1 with IFT88. We show that RUVBL1 orthologues have an asymmetric left-sided distribution at both the mouse embryonic node and the Kupffer’s vesicle in zebrafish embryos, with the latter asymmetry dependent on DNAAF1. These results suggest that DNAAF1-RUVBL1 biochemical and genetic interactions have a novel functional role in symmetry breaking and cardiac development