🔢Algebraic K-Theory Unit 2 – Categories and Functors

Key Concepts and Definitions

• Categories consist of objects and morphisms between those objects
• Morphisms are arrows that represent structure-preserving maps between objects
• Identity morphisms map an object to itself and satisfy left and right identity laws
• Composition of morphisms is associative and satisfies the composition law
  • If $f: A \rightarrow B$ and $g: B \rightarrow C$, then $g \circ f: A \rightarrow C$
• Isomorphisms are morphisms with two-sided inverses
• Functors are structure-preserving maps between categories
  • Functors map objects to objects and morphisms to morphisms
• Natural transformations are morphisms between functors

Historical Context and Development

• Category theory emerged in the 1940s from algebraic topology and homological algebra
• Saunders Mac Lane and Samuel Eilenberg introduced categories, functors, and natural transformations
• Alexander Grothendieck applied category theory to algebraic geometry in the 1950s and 1960s
• Algebraic K-theory, introduced by Daniel Quillen in the 1970s, heavily relies on category theory
• Quillen's Q-construction and the plus construction are foundational in algebraic K-theory
• Friedhelm Waldhausen's work in the 1980s further developed algebraic K-theory using category theory
• Modern algebraic K-theory continues to utilize and advance categorical concepts and techniques

Category Theory Fundamentals

• Small categories have sets of objects and morphisms, while large categories may have proper classes
• Opposite categories $C^{op}$ reverse the direction of morphisms
• Product categories $C \times D$ consist of pairs of objects and morphisms from $C$ and $D$
• Initial objects have exactly one morphism to every object in the category
• Terminal objects have exactly one morphism from every object in the category
• Pullbacks and pushouts are universal constructions that generalize fibers and cofibers
  • Pullbacks are limits of diagrams of the form $A \rightarrow C \leftarrow B$
  • Pushouts are colimits of diagrams of the form $A \leftarrow C \rightarrow B$
• Adjoint functors are pairs of functors $(F, G)$ with a natural isomorphism $\text{Hom}(FA, B) \cong \text{Hom}(A, GB)$

Functor Types and Properties

• Covariant functors preserve the direction of morphisms
• Contravariant functors reverse the direction of morphisms
• Faithful functors inject morphisms between any two objects
• Full functors surject morphisms between any two objects
• Essentially surjective functors map objects up to isomorphism
• Equivalences of categories are full, faithful, and essentially surjective functors
  • Equivalences preserve all categorical properties
• Adjunctions are pairs of functors $(F, G)$ with unit and counit natural transformations satisfying triangle identities
• Monads are endofunctors $T: C \rightarrow C$ with unit and multiplication natural transformations satisfying associativity and unit laws

Applications in Algebraic K-Theory

• Algebraic K-theory studies invariants of rings, schemes, and categories using functors and homotopy theory
• The K-theory functor $K: \text{Rings} \rightarrow \text{Spectra}$ assigns a spectrum to each ring
• Higher K-groups $K_n(R)$ are the homotopy groups of the K-theory spectrum $K(R)$
• The Q-construction $Q(C)$ is a category that models the K-theory space of a category $C$
• The plus construction $X \mapsto X^+$ is a functor that abelianizes the fundamental group of a space $X$
• The Waldhausen S-construction $wS_{\bullet}(C)$ is a simplicial category that models the K-theory of a Waldhausen category $C$
• The Thomason-Trobaugh construction extends algebraic K-theory to schemes using perfect complexes

Connections to Other Mathematical Fields

• Algebraic topology: K-theory originated from the study of vector bundles and topological K-theory
• Algebraic geometry: K-theory of schemes and coherent sheaves is a powerful invariant
• Representation theory: K-theory of group rings and equivariant K-theory relate to representation theory
• Number theory: K-theory of number fields and rings of integers connects to class field theory and L-functions
• Noncommutative geometry: K-theory is a key tool in the study of noncommutative spaces and $C^*$-algebras
• Homotopy theory: K-theory is a generalized cohomology theory and relates to stable homotopy theory
• Mathematical physics: K-theory appears in string theory, conformal field theory, and topological phases of matter

Problem-Solving Techniques

• Diagram chasing: Prove statements by following elements through commutative diagrams
• Universal properties: Utilize the uniqueness of morphisms guaranteed by universal constructions
• Yoneda lemma: Embed a category into a category of functors to study its properties
  • The Yoneda embedding $\text{Hom}(-, A): C \rightarrow [C^{op}, \text{Set}]$ is fully faithful
• Spectral sequences: Compute K-groups using convergent spectral sequences arising from filtrations or towers
• Localization and completion: Study K-theory by inverting or completing at certain classes of morphisms
• Descent and Mayer-Vietoris: Compute K-theory of global objects from local data using descent spectral sequences
• Waldhausen additivity: Split K-theory computations using categorical sum decompositions

Advanced Topics and Current Research

• Brave new algebra studies ring spectra and categories enriched over spectra
• Motivic homotopy theory combines algebraic geometry and stable homotopy theory
  • Motivic cohomology and motivic K-theory are active areas of research
• Equivariant K-theory studies K-theory of spaces and rings with group actions
• K-theory of derived categories and triangulated categories extends K-theory to more general settings
• Trace methods relate K-theory to cyclic homology and topological cyclic homology
• K-regularity and the Vorst conjecture study when K-theory and G-theory agree
• Noncommutative motives and K-theory of dg categories are recent developments in noncommutative geometry