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