See: Math notebook, Category theory glossary

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

Questions

**Six operations**

- the direct image {$f_*:Sh(X) → Sh(Y)$}
- the inverse image (or pullback sheaf) {$f^*:Sh(Y) → Sh(X)$}
- the proper (or extraordinary) direct image {$f_!:Sh(X) → Sh(Y)$}
- the proper (or extraordinary) inverse image {$f^!:D(Sh(Y)) → D(Sh(X))$}
- internal tensor product
- internal Hom

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**

- Wikipedia: Six operations
- Fritz Hoermann
- Cohomology theories in motivic stable homotopy theory
- What unifies stable homotopy theory and six functors
- Triangulated Categories of Mixed Motives
- The six operations in equivariant motivic homotopy theory Marc Hoyois
- Quantization via Linear homotopy types, Urs Schreiber
- Isomorphisms between left and right adjoints
- Dennis Gaitsgory, Notes on Geometric Langlands

**Videos**

Parsiųstas iš http://www.ms.lt/sodas/Book/GrothendieckSixOperations

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