Shai Keidar

This is a joint work with Tomer Schlank.

In this work we define formally the notion of an "asymptotically defined natural transformation", and use the periodicity theorem to construct one on the category of compact spectra of type n. Inverting it, we get the category of compact T(n)-local spectra. We show that any category with an asymptotically defined natural isomorphism (in particular compact T(n)-local spectra) admits a coherent group action and use it to show there is an embedding of a large group in the automorphism of this category, which are identified with the telescopic Picard group.

Utilizing Kummer theory we construct a lift of a non-abelian Galois extensions from the K(n)-local category to the T(n)-local categry.

We show that in a presentably symmetric monoidal category C, if G is a C-dualizable group then faithful Galois extensions are co-representable: A faitfhul Galois extension in C is the same data as a symmetric monoidal functor from Spc^{BG} to C.

Using this we show that for nice semiadditive categories of height n there is an pro-(n+1)-finite Galois space classifiying n-finite Galois extensions, extending a result of Mathew for stable categories. We also extend Kummer theory to this setting, and discuss Galois-closure of categories.

Using the co-representability of Galois extensions (see above), we prove that for a semiadditive category C, the data of a cyclic Galois extension and a strict unit is equivalent to a symmetric monoidal, colimit preserving functor from Mod_{Z}(C)^{B Z/m} to C.

Given such a symmetric monoidal functor F, we define its corresponding cyclic Azumaya algebra as F(Z/m⋉Z) extending the constructions of Baker-Richter-Syzmik and Mor.

We use it to construct a cyclic subgroup of order p-1 in the telescopic Brauer group.

This is a joint work with Shauly Ragimov.

In this work we extend the construction of the alternating power of a module to the higher semiadditive settings, where the symmetric group admits many more characters (even when p=2). Similarly to how alternating and symmetric powers decategorify to power operations on KU, we show that our alternating powers decategorify to "twisted semiadditive" power operations.

For categories with cyclotomically closed unit of height n, characters of the (n+1)-st stable stem induce a character on all symmetric groups. We show that for these characters all alternating powers assemble to an algebra, generalizing the classical symmetric and exterior algebras.

Using iterative application of the induced character formula of Carmeli-Cnossen-Ramzi-Yanvoski we prove a height induction formula for two categories of interest and use it to compute the monoidal dimensions of alternating powers in low heights.

This is a joint work with Leor Neuhause, Tomer Schlank, Lior Yanvoski.

We define, using the Gray tensor product, the lax external product of a d-monoidal (∞,∞)-category and an e-monoidal (∞,∞)-category, returning and (d+e)-monoidal (∞,∞)-category.
Denoting by FD(n,d) the free d-monoidal category generated by an n-dualizable object we show that the lax external product of FD(n,d) and FD(m,e) identifies with FD(n+m,d+e).

In particular we show that FD(∞,∞) is an idempotent algebra with respect to a lax tensor product and assuming the cobordism hypothesis get a multiplicative-structure Pontryagin-Thom isomorphism.

In a work in progress we construct maps from the lax external product of framed tangle categories to an appropriate tangle category. Thus reformulating the Tangle hypothesis to proving that these maps are isomorphisms.