Intro Category Theory Talk

Tags: #todo #projects/my-talks #higher-algebra/category-theory

Intro Category Theory Talk

  • Definition: category (objects, morphisms, composition)
  • Big list of examples
    • \({\mathsf{Grp}}, {\mathsf{R}{\hbox{-}}\mathsf{Mod}}, {\mathsf{Alg}_{/k} }, {\mathsf{Top}}, \mathsf{Loc}\mathsf{RingSp}, [\mathsf{C}, \mathsf{D}]\),
    • The free category on a poset
    • \(\FinSet_+\), the category of finite linearly ordered sets with objects of the form \([n] = \left\{{0, \cdots, n}\right\}\).
    • \(\Delta\) the simplex category : finite totally ordered sets.
    • The groupoid associated to a group, general groupoids
    • \({\mathsf{Grpd}}, \mathsf{Cat}\)
    • \({\mathsf{Open}}(X)\) for \(X\in {\mathsf{Top}}\)
    • \({\mathsf{sm}}{\mathsf{Mfd}}\)
    • \({\mathsf{Sch}}\), ${\mathsf{Sch}}_{/ {S}} $
  • Definition: functor
    • Examples:
      • \(\pi_1: {\mathsf{Top}}\to {\mathsf{Grp}}\)
      • \(\pi_*: {\mathsf{Top}}\to {\mathsf{gr}\,}_{\mathbb{Z}}{\mathsf{Grp}}\)
  • Definition: Natural transformations
    • Interpretation as morphisms in \([\mathsf{C}, \mathsf{D}]\).
  • Definition: isomorphism of objects
  • Equivalence of categories:
  • Definition: adjunction
    • Definition: unit of an adjunction and counit
    • Examples:
      • tensor-hom adjunction
      • free-forgetful adjunction
      • Frobenius reciprocity
      • Cartesian closed category
      • restriction and extension of scalars adjunction
      • Definition: adjoint equivalence
      • adjoint functor theorem
  • Useful constructions
    • Initial and terminal objects
    • Universal properties
      • Quotient group or quotient topology
      • Tensor product of modules
    • Product and coproduct
    • Pullback and pushout
    • slice category and under category
      • cone category
    • Colimit and limit
      • As initial/terminal objects in cone category
      • RAPL and LAPC
    • coequalizer and equalizer
  • Definition: presheaf and sheaf
  • Definition: representable functor
  • Definition: Yoneda embedding
    • Philosophy: functor of points

Further Topics

  • reflective and coreflective subcategories
  • cocontinuous functor and continuity of \(\mathop{\mathrm{Hom}}\).
  • cocomplete category
  • monad, algebra over a monad, and Beck's monadicity theorem
  • monoid object
  • monoidal category
  • cofinal functor
  • filtered category and filtered colimits
#todo #projects/my-talks #higher-algebra/category-theory