This is an expository introduction to simplicial sets and simplicial homotopy theory with particular focus on relating the combinatorial aspects of the theory to their geometric/topological origins. It is intended to be accessible to students familiar with just the fundamentals of algebraic topology.
We construct a weak representation of the category of framed affine tangles on a disjoint union of triangulated categories ${mathcal D}_{2n}$. The categories we use are that of coherent sheaves on Springer fibers over a nilpotent element of $sl_{2n}$ with two equal Jordan blocks. This representation allows us to enumerate the irreducible objects in the heart of the exotic $t$-structure on ${mathcal D}_{2n}$ by crossingless matchings of $2n$ points on a circle. We also describe the algebra of endomorphisms of the direct sum of the irreducible objects.
We give a concrete description of the category of etale algebras over the ring of Witt vectors of a given finite length with entries in an arbitrary ring. We do this not only for the classical p-typical and big Witt vector functors but also for variants of these functors which are in a certain sense their analogues over arbitrary local and global fields. The basic theory of these generalized Witt vectors is developed from the point of view of commuting Frobenius lifts and their universal properties, which is a new approach even for the classical Witt vectors. The larger purpose of this paper is to provide the affine foundations for the algebraic geometry of generalized Witt schemes and arithmetic jet spaces. So the basics here are developed somewhat fully, with an eye toward future applications.
The results of Iv. Prodanov on abstract spectra and separative algebras were announced in the journal Trudy Mat. Inst. Steklova, 154, 1983, 200--208, but their proofs were never written by him in the form of a manuscript, preprint or paper. Since the untimely death of Ivan Prodanov withheld him from preparing the full version of this paper and since, in our opinion, it contains interesting and important results, we undertook the task of writing a full version of it and thus making the results from it known to the mathematical community. So, the aim of this paper is to supply with proofs the results of Ivan Prodanov announced in the cited above paper, but we added also a small amount of new results. The full responsibility for the correctness of the proofs of the assertions presented below in this work is taken by us; just for this reason our names appear as authors of the present paper.
We study idempotents in intensional Martin-Lof type theory, and in particular the question of when and whether they split. We show that in the presence of propositional truncation and Voevodskys univalence axiom, there exist idempotents that do not split; thus in plain MLTT not all idempotents can be proven to split. On the other hand, assuming only function extensionality, an idempotent can be split if and only if its witness of idempotency satisfies one extra coherence condition. Both proofs are inspired by parallel results of Lurie in higher category theory, showing that ideas from higher category theory and homotopy theory can have applications even in ordinary MLTT. Finally, we show that although the witness of idempotency can be recovered from a splitting, the one extra coherence condition cannot in general; and we construct the type of fully coherent idempotents, by splitting an idempotent on the type of partially coherent ones. Our results have been formally verified in the proof assistant Coq.
We give an exposition and generalization of Orlovs theorem on graded Gorenstein rings. We show the theorem holds for non-negatively graded rings which are Gorenstein in an appropriate sense and whose degree zero component is an arbitrary non-commutative right noetherian ring of finite global dimension. A short treatment of some foundations for local cohomology and Grothendieck duality at this level of generality is given in order to prove the theorem. As an application we give an equivalence of the derived category of a commutative complete intersection with the homotopy category of graded matrix factorizations over a related ring.
We introduce a notion of bimodule in the setting of enriched $infty$-categories, and use this to construct a double $infty$-category of enriched $infty$-categories where the two kinds of 1-morphisms are functors and bimodules. We then consider a natural definition of natural transformations in this context, and show that in the underlying $(infty,2)$-category of enriched $infty$-categories with functors as 1-morphisms the 2-morphisms are given by natural transformations.
We generalize toposic Galois theory to higher topoi. We show that locally constant sheaves in a locally (n-1)-connected n-topos are equivalent to representations of its fundamental pro-n-groupoid, and that the latter can be described in terms of Galois torsors. We also show that finite locally constant sheaves in an arbitrary infinity-topos are equivalent to finite representations of its fundamental pro-infinity-groupoid. Finally, we relate the fundamental pro-infinity-groupoid of 1-topoi to the construction of Artin and Mazur and, in the case of the etale topos of a scheme, to its refinement by Friedlander.
Logicians and philosophers of science have proposed various formal criteria for theoretical equivalence. In this paper, we examine two such proposals: definitional equivalence and categorical equivalence. In order to show precisely how these two well-known criteria are related to one another, we investigate an intermediate criterion called Morita equivalence.
We describe an abstract 2-categorical setting to study various notions of polynomial and analytic functors and monads.
