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Introduction E9F5FC

Understandable FFFFFF

Questions FFFFC0

Notes EEEEEE

Software

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")