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Six operations

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 {$f^!:D(Sh(Y)) → D(Sh(X))$}

Abelian sheaves

• internal tensor product {$\bigotimes^L:C^{op}\times C\rightarrow C$}
• internal Hom {$\textrm{RHom}:C^{op}\times C\rightarrow C$}

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.
• The exceptional inverse image functor ({$Rf^!$} or {$f^!$}?) is the fourth and most sophisticated in a series of image functors for sheaves. It is needed to express Verdier duality in its most general form.
• Verdier duality exchanges "∗" and "!".
  • Verdier duality states that certain image functors for sheaves are actually adjoint functors.
  • 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).
  • It is an analog for locally compact spaces of the coherent duality for schemes due to Alexander Grothendieck.
  • It is commonly encountered when studying constructible or perverse sheaves.
• Categories that possess an internal Hom functor are referred to as closed categories. One has that

{$\text{Hom}(I,\text{hom}(-,-))\simeq \text{Hom}(-,-)$}

where {$I$} is the unit object of the closed category. For the case of a closed monoidal category, this extends to the notion of currying, namely, that

{$\text{Hom}(X,Y\Rightarrow Z)\simeq \text{Hom}(X\otimes Y,Z)$}

where {$\otimes$} is a bifunctor, the internal product functor defining a monoidal category. The isomorphism is natural in both {$X$} and {$Z$}. In other words, in a closed monoidal category, the internal Hom functor is an adjoint functor to the internal product functor. The object {$Y\Rightarrow Z$} is called the internal {$\textrm{Hom}$}. When {$\otimes$} is the Cartesian product {$\times$} , the object {$Y\Rightarrow Z$} is called the exponential object, and is often written as {$Z^{Y}$}.

• Internal Homs, when chained together, form a language, called the internal language of the category. The most famous of these are simply typed lambda calculus, which is the internal language of Cartesian closed categories, and the linear type system, which is the internal language of closed symmetric monoidal categories.
• Grothendieck's six operations express the imagination. They arise in representations and circumstances, where there is a perspective on a perspective, in the chain of human and divine perspectives: HDHDH. Anything and Nothing go in the same direction, as do Everything and Something. Thus negations go in the same direction between human and divine perspectives.
• The adjunction, the left adjoint and right adjoint, together open up from D and then (by way of C) close back down into D. So we may think of a fourfold interpretation of C (by sheaf functors?) and the two functors (tensor and homset) of leaving from D and going to D.

Six bases of the symmetric functions

• Perhaps the elementary and the homogeneous relate to the Internal hom and the Tensor product.
• The four other bases range from simplest to complex: monomial, power, Schur, forgotten.
• The forgotten basis models the gaps in the system, as with the prime numbers. How do the cycles that ground the forgotten basis relate to those in number theory?

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")

读物

• Olivia Caramello. Derived categories, derived functors and Grothendieck's six operations
• Martin Gallauer. An introduction to six-functor formalims.
• Borcherd's videos on Algebra Geometry II (Sheaves) relating {$f_*$} and {$f^{-1}$} and relating {$f_*$} and {$f^*$}.
• Wikipedia: Six operations
  • Wikipedia: Image functors for sheaves
  • Six operations at nLab
  • Six operations at Math Stack Exchange
• Fausk, Hu, May. Isomorphisms Between Left and Right Adjoints, Halvard Fausk
• Fritz Hoermann
  • Fibered Derivators, (Co)homological descent, and Grothendieck's six functors
  • Six Functor Formalisms and Fibered Multiderivators
  • Derivator Six Functor Formalisms - Definition and Construction I
• 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
• Drew, Gallauer. The Universal Six-Functor Formalism.
• Milano Summer School. The Six-Functor Formalism and Motivic Homotopy Theory.
• Deglise, Jin, Khan. Fundamental Classes in Motivic Homotopy Theory.
• Math Stack Exchange. Why is Voevodsky's motivic homotopy theory the right approach?
• Shanna Dobson, Chris Fields. Making Up Our Minds: Imaginative Deconstruction in MathArt, 1920 – Present

影片

• Deglise. Six-functors formalism in motivic homotopy type theory.
• Motivic Verona: minicourse on abstract and motivic homotopy theory by Peter Arndt
• Peter Arndt - Abstract motivic homotopy theory
• A p-Adic 6-Functor Formalism in Rigid-Analytic Geometry

Hypergraphs

• Wolfram on hypergraphs
• 维基百科: Hypergraph

Notes

• Six operations (six modeling methods) 6=4+2 relates 2 perspectives (internal (tensor) & external (Hom)) by 4 scopes (functors). The six 3+3 specifications define a gap for a perspective, thus relate three perspectives. Algebra and analysis come together in one perspective. Together, in the House of Knowledge, these all make for consciousness.

Unifying Themes in Geometry

• Laurent Lafforgue: “Geometry according to Grothendieck : glimpses from “Récoltes et Semailles””

Laurent Lafforgue about Grothendieck

• 21:00 The three aspects of thinking mathematically, of seeing mathematics: "le nombre" (number - arithmetic), "la grandeur" (size, quantity, magnitude), "la forme" (form, geometric). What fascinates him is form and, among its thousand faces, structure.
• 22:00 Structures cannot be invented. They are waiting for us to discover them, to grasp and understand them (apprehend them), to express them. To uncover them we have to invent and refine languages, build theories to account for what has been seen and understood. The quality of our inventiveness and imagination is the quality of the attention we pay, of our listening. (Compare with QTF.)
• Arithmetic (science of discrete structures), analysis (science of continuous structures), geometry straddles these two types of structure. Geometry is a land of infinite luxuriant riches which readily multiply infinitely under our hand. New geometry is a synthesis between
  • the arithmetic world of "spaces without continuity principle"
  • the world of continuous quantity
• Continuous and discrete in mathematical meaning and natural language meaning. (Three-cycle relates movement from discrete (induction) to continuous (limit), and from continuous (whole) to discrete (list)).
  • Invention: Continuity is what we have to give in our work. (Compare with three-cycle, with invention of a theoretical language.)
  • Discovery: Discontinuity is unexpected discovery given to us. (Compare with ways of figuring things out.)
• Six operations and biduality applies to both continuous coefficients and discrete coefficients.
• Cohomology of the crystalline site, with coefficients in the structure sheaf, identifies with De Rham cohomology.
• 35:00 Topos theory serves both discrete and continuous.
• Look for ubiquitous phenomena that get expressed both through coefficient coefficients and discrete coefficients.
  • Example "duality". Poincare duality for singular homology, Serre duality for coherent modules. Systems of six operations formalism, notion of constructible objects, biduality.
  • Grothendieck - Riemman-Roch for continuous coefficients, discrete coefficients.
  • Bridges between the discrete and continuous worlds. About the work of Zoghman Mebkhout (Riemman-Hilbert correspondence) which realizes a bridge between topology, algebra and analysis.
  • Weyl conjuectures - tight relationship between arithmetic and topology. Construction of cohomology invariants. Trace formulas for these invariants. "Yoga of weights".
  • Riemann-Hilbert correspondence as an equivalence between (linear) derived categories of differential nature (systems of linear PDE's) and of discrete nature.
  • Anabelian geometry program based on the key fact that objects of purely arithmetic nature {$\textrm{Gal}(\mathbb{Q})=\textrm{Aut}(\mathbb{Q}/\mathbb{Q})$} naturally act on objects of purely topological nature.
  • Bridges that consist in objects or theories that can specalize or localize both in the direction of the discrete and the direcction of the continuous (orthogonal polynomials and their weight functions).
    • Scheme theory over {$\textrm{Spec}(\mathbb{Z})$} with its natural base change functors to discrete geometry over finite fields, continuous geometry over R or C.
    • The theory of sites (small category C, Grothendieck topology J on C) which specializes to both (continuous) topological spaces and (discrete) small categories and allows intermediate definitions such as etale sites, crystalline sites...
    • Conjectural theory of motives that should specialize both to cohomology theories with discrete coefficients such as l-adic or singular cohomology, and Hodge cohomology, which is subject to continuous variations.
• Topos has a double definition.
  • Constructive definition as categories that can be (geometrically) presented as categories of sheaves on some site (C,J).
  • Axiomatic characterization (Giraud's theorem) as categories which share with Set the same list of constructive categorical properties.
• 1:40 Six functor formalism.
  • 1:47 Constructible object.

In Grothendieck's six operations, in the adjoint string for {$f$}, consider the special case when {$f$} is constant. And relate the functors to the representations of the nullsome.

• significant {$\forall$}
• constant
• direct {$\exists$}
• true ?

Grothendieck's six operations

• 2 conceptions (inner-outer)(algebra?): List = tensor (answer?) - homset (question?)
• 4 conceptions (analysis?): 4 sheave functors
• How SU(2) transformations 4+2 (normal form) relate to Grothendieck's 6 operations?

Understand how math expresses the six criteria (the six representations of divisions).

https://mathoverflow.net/questions/170319/what-if-anything-unifies-stable-homotopy-theory-and-grothendiecks-six-functor