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数学

Discovery

Andrius Kulikauskas

  • ms@ms.lt
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

Lietuvių kalba

Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software


See: Math notebook, Category theory glossary

Investigation: Specify the category theory that grounds Grothendieck's six operations.


Questions


Six operations

Let f: X → Y be a continuous mapping of topological spaces.

  • Sh(–) the category of sheaves of abelian groups on a topological space.
  • {$f_*$} generalizes the notion of a section of a sheaf to the relative case.
  • {$f^{-1}$} is the left adjoint of {$f^*$}.
  • {$f_*$} is right adjoint to {$f^*$}.
  • {$f_!$} and {$f^!$} form an adjoint functor pair.
  • {$f_!$} is an image functor for sheaves.
  • internal tensor product is left adjoint to internal Hom.
  • Verdier duality exchanges "∗" and "!". It is a generalization of Poincare duality, which says that the kth homology group is isomorphic to the (n-k)th homology group of an n-dimensional oriented closed manifold M (compact and without boundary).

Categorial foundations

  • A category is closed if it has an internal Hom functor.
  • A monoidal category, also called a tensor category, is a category C equipped with (1) a bifunctor {$\otimes: C \times C \to C$}, (2) an identity object and (3) natural isomorphisms that make ⊗ associative and the identity object an identity for ⊗, subject to certain coherence conditions.
  • A category is abelian if it has a zero object, it has all pullbacks and pushouts, and all monomorphisms and epimorphisms are normal.

Concepts

  • "Continuous" and "discrete" duality (derived categories and "six operations")

Readings

Videos

GrothendieckSixOperations


Naujausi pakeitimai


Puslapis paskutinį kartą pakeistas 2019 balandžio 06 d., 12:50
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