Abstract representation theory of Dynkin quivers of type A

We study the representation theory of Dynkin quivers of type A in abstract stable homotopy theories, including those associated to fields, rings, schemes, differential-graded algebras, and ring spectra. Reflection functors, (partial) Coxeter functors, and Serre functors are defined in this generality and these equivalences are shown to be induced by universal tilting modules, certain explicitly constructed spectral bimodules. In fact, these universal tilting modules are spectral refinements of classical tilting complexes. As a consequence we obtain split epimorphisms from the spectral Picard groupoid to derived Picard groupoids over arbitrary fields. These results are consequences of a more general calculus of spectral bimodules and admissible morphisms of stable derivators. As further applications of this calculus we obtain examples of universal tilting modules which are new even in the context of representations over a field. This includes Yoneda bimodules on mesh categories which encode all the other universal tilting modules and which lead to a spectral Serre duality result. Finally, using abstract representation theory of linearly oriented $A_n$-quivers, we construct canonical higher triangulations in stable derivators and hence, a posteriori, in stable model categories and stable $\infty$-categories

Associative nn-categories

We define novel fully combinatorial models of higher categories. Our definitions are based on a connection of higher categories to "directed spaces". Directed spaces are locally modelled on manifold diagrams, which are stratifications of the n-cube such that strata are transversal to the flag foliation of the n-cube. The first part of this thesis develops a combinatorial model for manifold diagrams called singular n-cubes. In the second part we apply this model to build our notions of higher categories. Singular n-cubes are "directed triangulations" of space together with a decomposition into a collection of subspaces or strata. Singular n-cubes can be naturally organised into two categories. The first, whose morphisms are bundles themselves, is used for the inductive definition of singular (n+1)-cubes. The second, whose morphisms are "open" base changes, admits an (epi,mono) factorisation system. Monomorphisms will be called embeddings of cubes. Epimorphisms will be called collapses and describe how triangulations can be coarsened. Each cube has a unique coarsest triangulation called its normal form. The existence of normal forms makes the equality relation of (combinatorially represented) manifold diagrams decidable. As the main application of the resulting combinatorial framework for manifold diagrams, we give algebraic definitions of various notions of higher categories. Namely, we define associative n-categories, presented associative n-categories and presented associative n-groupoids. All three notions will have strict units and associators; the only weak coherences are homotopies, but we develop a mechanism for recovering the usual coherence data of weak n-categories, such as associators and pentagonators and their higher analogues. This will motivate the conjecture that the theory of associative higher categories is equivalent to its fully weak counterpart.Comment: 499 page

CMCS 2010 Categorifying Computations into Components via Arrows as Profunctors

The notion of arrow by Hughes is an axiomatization of the algebraic structure possessed by structured computations in general. We claim that an arrow also serves as a basic component calculus for composing state-based systems as components—in fact, it is a categorified version of arrow that does so. In this paper, following the second author’s previous work with Heunen, Jacobs and Sokolova, we prove that a certain coalgebraic modeling of components—which generalizes Barbosa’s—indeed carries such arrow structure. Our coalgebraic modeling of components is parametrized by an arrow A that specifies computational structure exhibited by components; it turns out that it is this arrow structure of A that is lifted and realizes the (categorified) arrow structure on components. The lifting is described using the first author’s recent characterization of an arrow as an internal strong monad in Prof, the bicategory of small categories and profunctors

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems