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Register a new userThe AKSZ Construction in Derived Algebraic Geometry as an Extended Topological Field Theory
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We construct a family of oriented extended topological field theories using the AKSZ construction in derived algebraic geometry, which can be viewed as an algebraic and topological version of the classical AKSZ field theories that occur in physics. These have as their targets higher categories of symplectic derived stacks, with higher morphisms given by iterated Lagrangian correspondences. We define these, as well as analogous higher categories of oriented derived stacks and iterated oriented cospans, and prove that all objects are fully dualizable. Then we set up a functorial version of the AKSZ construction, first implemented in this context by Pantev-Toen-Vaquie-Vezzosi, and show that it induces a family of symmetric monoidal functors from oriented stacks to symplectic stacks. Finally, we construct forgetful functors from the unoriented bordism $(infty,n)$-category to cospans of spaces, and from the oriented bordism $(infty,n)$-category to cospans of spaces equipped with an orientation; the latter combines with the AKSZ functors by viewing spaces as constant stacks, giving the desired field theories.
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We consider generalized $Lambda$-structures on algebras and schemes over the ring of integers $mathit{O}_K$ of a number field $K$. When $K=mathbb{Q}$, these agree with the $lambda$-ring structures of algebraic K-theory. We then study reduced finite flat $Lambda$-rings over $mathit{O}_K$ and show that the maximal ones are classified in a Galois theoretic manner by the ray class monoid of Deligne and Ribet. Second, we show that the periodic loci on any $Lambda$-scheme of finite type over $mathit{O}_K$ generate a canonical family of abelian extensions of $K$. This raises the possibility that $Lambda$-schemes could provide a framework for explicit class field theory, and we show that the classical explicit class field theories for the rational numbers and imaginary quadratic fields can be set naturally in this framework. This approach has the further merit of allowing for some precise questions in the spirit of Hilberts 12th Problem. In an interlude which might be of independent interest, we define rings of periodic big Witt vectors and relate them to the global class field theoretical mathematics of the rest of the paper.
We present an introduction to the theory of algebraic geometry codes. Starting from evaluation codes and codes from order and weight functions, special attention is given to one-point codes and, in particular, to the family of Castle codes.
Structures where we have both a contravariant (pullback) and a covariant (pushforward) functoriality that satisfy base change can be encoded by functors out of ($infty$-)categories of spans (or correspondences). In this paper we study the more complicated setup where we have two pushforwards (an additive and a multiplicative one), satisfying a distributivity relation. Such structures can be described in terms of bispans (or polynomial diagrams). We show that there exist $(infty,2)$-categories of bispans, characterized by a universal property: they corepresent functors out of $infty$-categories of spans where the pullbacks have left adjoints and certain canonical 2-morphisms (encoding base change and distributivity) are invertible. This gives a universal way to obtain functors from bispans, which amounts to upgrading monoid-like structures to ring-like ones. For example, symmetric monoidal $infty$-categories can be described as product-preserving functors from spans of finite sets, and if the tensor product is compatible with finite coproducts our universal property gives the canonical semiring structure using the coproduct and tensor product. More interestingly, we encode the additive and multiplicative transfers on equivariant spectra as a functor from bispans in finite $G$-sets, extend the norms for finite etale maps in motivic spectra to a functor from certain bispans in schemes, and make $mathrm{Perf}(X)$ for $X$ spectral Deligne--Mumford stack a functor of bispans using a multiplicative pushforward for finite etale maps in addition to the usual pullback and pushforward maps.
This book is an introduction to 2-categories and bicategories, assuming only the most elementary aspects of category theory. A review of basic category theory is followed by a systematic discussion of 2-/bicategories, pasting diagrams, lax functors, 2-/bilimits, the Duskin nerve, 2-nerve, adjunctions and monads in bicategories, 2-monads, biequivalences, the Bicategorical Yoneda Lemma, and the Coherence Theorem for bicategories. Grothendieck fibrations and the Grothendieck construction are discussed next, followed by tricategories, monoidal bicategories, the Gray tensor product, and double categories. Completely detailed proofs of several fundamental but hard-to-find results are presented for the first time. With exercises and plenty of motivation and explanation, this book is useful for both beginners and experts.
By a ring groupoid we mean an animated ring whose i-th homotopy groups are zero for all i>1. In this expository note we give an elementary treatment of the (2,1)-category of ring groupoids (i.e., without referring to general animated rings and without using n-categories for n>2). The note is motivated by the fact that ring stacks play a central role in the Bhatt-Lurie approach to prismatic cohomology.
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