During the post-Darwinian history of taxonomy, the Linnaean hierarchy has maintained its role as a means for representing hierarchical taxonomic relationships. During the same period, the principle of descent has taken on an increasingly important role as the basis for reformulated versions of fundamental taxonomic concepts and principles. Early in this history, the principle of descent provide...

We introduce flagged (∞, n)-categories and prove that they are equivalent to Segal sheaves on Joyal’s category Θn. As such, flagged (∞, n)-categories provide a model-independent formulation of Segal sheaves. This result generalizes the statement that n-groupoid objects in spaces are effective, as we explain and contextualize. Along the way, we establish a useful expression for the univalent-com...

During the post-Darwinian history of taxonomy, the Linnaeän hierarchy has maintained its role as a means for representing hierarchical taxononaic relationships. During the same period, the principle of descent has tallen on an increasingly important role as the basis for reformulated versions of fundamental taxonomic concepts and principles. Early in this history, the principle of descent provi...

We give a homotopy theoretic characterization of sheaves on a stack and, more generally, a presheaf of groupoids on an arbitary small site C. We use this to prove homotopy invariance and generalized descent statements for categories of sheaves and quasi-coherent sheaves. As a corollary we obtain an alternate proof of a generalized change of rings theorem of Hovey.

Given a pseudomonad $mathcal{T} $ on a $2$-category $mathfrak{B} $, if a right biadjoint $mathfrak{A}tomathfrak{B} $ has a lifting to the pseudoalgebras $mathfrak{A}tomathsf{Ps}textrm{-}mathcal{T}textrm{-}mathsf{Alg} $ then this lifting is also right biadjoint provided that $mathfrak{A} $ has codescent objects. In this paper, we give  general results on lifting of biadjoints. As a consequence, ...

The Holy Quran was descended from Almighty Allah to the Holy Prophet (AS) about the last 14 centuries and there has been a time interval between descent of the Quran and its existing readers and commentators. This Question is posed whether this time interval between descent of the Quran and commentator will effect on verses understanding and their commentary or not? As one of commentary princip...

We prove descent theorems for semiorthogonal decompositions using techniques from derived algebraic geometry. Our methods allow us to capture more general filtrations of categories and even marked filtrations, where one descends not only admissible subcategories but also preferred objects.

In this article, we develop a theory of Grothendieck’s six operations for derived categories in étale cohomology of Artin stacks. We prove several desired properties of the operations, including the base change theorem in derived categories. This extends all previous theories on this subject, including the recent one developed by Laszlo and Olsson, in which the operations are subject to more as...