Category Theory
Model Categories
Daniel Quillen in 1967 axiomatized homotopy theory in 'Homotopical Algebra', creating a language spanning topology, algebraic geometry, and type theory. Model categories are the skeleton of modern homotopical mathematics.
- Derived algebraic geometry: infinity-toposes are built as model categories of simplicial sheaves
- Type theory: model structures on type-theoretic languages give semantics for HoTT
- Stable homotopy: spectra (models for generalized cohomology theories) live in model categories
Предварительные знания
Model Category Axioms
Quillen introduced model categories in his 1967 monograph Homotopical Algebra to put homotopy theory and homological algebra under one roof. The framework now powers modern foundations: Lurie's 2009 book Higher Topos Theory builds quasicategories on Joyal-Quillen model structures, and Bousfield localization gives the chromatic stable homotopy categories used in the resolution of the Kervaire invariant problem (Hill-Hopkins-Ravenel, 2009).
A model category (C, W, Cof, Fib) satisfies MC1-MC5: (MC1) finite limits/colimits exist; (MC2) two-out-of-three for W; (MC3) closure under retracts; (MC4) ACof has LLP against Fib, Cof has LLP against AFib; (MC5) every morphism factors as (ACof, Fib) and (Cof, AFib). Ho(C) = C[W^{-1}] is the homotopy category.
What does axiom MC2 (two-out-of-three) state?
Examples and Applications
Canonical examples: (1) sSet with Kan-Quillen; (2) Ch(k) with quasi-isomorphisms -- model for the derived category D(k); (3) CGWH topological spaces, Quillen-equivalent to sSet via geometric realization and singular complex. Model equivalences preserve Ho(C) and mapping spaces Map(X,Y).
What model structure on Ch(k) reproduces the derived category D(k)?
Key Ideas
- Model category (C, W, Cof, Fib): MC1-MC5 include closure, lifting, factorization
- ACof = W cap Cof, AFib = W cap Fib -- the trivial classes
- Ho(C) = C[W^{-1}]: morphisms are homotopy classes QX -> RY
- SOA builds factorizations by transfinite composition of pushouts
- sSet ~Q Top: simplicial sets and topological spaces are Quillen equivalent
Further Directions
These ideas open paths to deeper mathematics.
- ct-29-homotopy-type-theory — extends
Вопросы для размышления
- Give a concrete example.
- How does this connect to other areas of mathematics?