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Online seminář z algebry - 21.5.2020 PDF Tisk

Další seminář z algebry se koná 21.5.2020 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Axel Osmond

Towards a 2-dimensional spectral construction

Many prominent dualities in mathematics are instances of a common con- struction centered on the notion of spectral functor. Roughly stated, one starts with a locally finitely presentable category, equipped with a  subcategory of distinguished local objects encoding point-like data and a factorization system (Etale maps, Local maps) where the etale maps behave as duals of distinguished continuous maps. Several manners of axiomatizing the correct relation between those ingredients have been proposed, either through topos theoretic methods by “localizing” local objects with a Grothendieck topology generated by etale maps, or in an alternative (though tightly related) way based on the notion of local right ajoint (or equivalently stable functor). Then the spectrum of a  given object is constructed as a topos classifying etale maps under this given object toward local objects, equipped with a structural sheaf playing the role of the “free local object” under it. This defines a  spectral functor from the ambient locally finitely presentable category to a category of locally structured toposes, forming an adjunction with a  corresponding global section functor.
This construction provide a convenient template for several prominent 1-catego- rical examples, as dualities for rings in algebraic geometry, or also Stone-like dualities for different classes of propositional algebras. The strong analogy between those dualities and their corresponding first order syntax-semantics dualities suggests the later could be understood as instances of a convenient 2- dimensional spectral construction. In this talk we will expose the ongoing work devoted to concretize this intuition.
After recalling the 1-dimensional version of the construction and the details of some prominent Stone-like examples, we introduces a notion of stable 2- functor and provide a method to construct an associated notion of spectral 2-sites, defining the spectrum as the associated Grothendieck 2-topos equiped with a distinguished structural stack. In particular we give a special interest in determining the local objects and the factorization system associated to doc- trines corresponding to fragments of first order logics, as Lex, Reg, or Coh; in those situations, the construction simplifies as the spectral site happens to be 1-truncated so that one recover the corresponding 1-dimensional notion of clas- sifying topos of a theory as the spectrum, and the geometry of the spectrum actually arises from the geometric properties of local toposes and etale geometric morphisms.

Aktualizováno Úterý, 19 Květen 2020 09:10
Online seminář z algebry - 14.5.2020 PDF Tisk

Další seminář z algebry se koná 14.5.2020 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Soichiro Fujii

A unified framework for notions of algebraic theory

Universal algebra uniformly captures various algebraic structures, by expressing them as equational theories or abstract clones. The ubiquity of algebraic structures in mathematics and related fields has given rise to several variants of universal algebra, such as theories of symmetric operads, non-symmetric operads, generalised operads, PROPs, PROs, and monads. These variants of universal algebra are called notions of algebraic theory. In this talk, we present a  unified framework for them. The key observation is that each notion of algebraic theory can be identified with a monoidal category, in such a way that algebraic theories correspond to monoid objects therein. To incorporate semantics, we introduce a categorical structure called metamodel, which formalises a  definition of models of algebraic theories. We also define morphisms between notions of algebraic theory, which are a  monoidal version of profunctors. Every strong monoidal functor gives rise to an adjoint pair of such morphisms, and provides a uniform method to establish isomorphisms between categories of models in different notions of algebraic theory. A general structuresemantics adjointness result and a double categorical universal property of categories of models are also shown.

Aktualizováno Úterý, 12 Květen 2020 09:02