Tags: #todo #my_talks #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 essentially surjective
  • Definition: full functor
  • Definition: faithful functor
  • Definition: equivalence of categories
  • Philosophy: isomorphism vs equivalence
• 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