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**►****Foundations****▼****Algebra****►****11-XX Number theory****▼****12-XX Field theory and polynomials****►****12-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****►****12Dxx Real and complex fields****►****12Exx General field theory****►****12Fxx Field extensions****►****12Gxx Homological methods (field theory)****►****12Hxx Differential and difference algebra****►****12Jxx Topological fields****►****12Kxx Generalizations of fields****►****12Lxx Connections with logic**- 12Y05 Computational aspects of field theory and polynomials

**►****13-XX Commutative rings and algebras****►****14-XX Algebraic geometry****►****15-XX Linear and multilinear algebra; matrix theory****►****16-XX Associative rings and algebras****►****17-XX Nonassociative rings and algebras****▼****18-XX Category theory; homological algebra****►****18-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****▼****18Axx General theory of categories and functors**- 18A05 Definitions, generalizations
- 18A10 Graphs, diagram schemes, precategories
- 18A15 Foundations, relations to logic and deductive systems
- 18A20 Epimorphisms, monomorphisms, special classes of morphisms, null morphisms
- 18A22 Special properties of functors (faithful, full, etc.)
- 18A23 Natural morphisms, dinatural morphisms
- 18A25 Functor categories, comma categories
- 18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
- 18A32 Factorization of morphisms, substructures, quotient structures, congruences, amalgams
- 18A35 Categories admitting limits (complete categories), functors preserving limits, completions
- 18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
- 18A99 None of the above, but in this section

**►****18Bxx Special categories****▼****18Cxx Categories and theories**- 18C05 Equational categories
- 18C10 Theories (e.g. algebraic theories), structure, and semantics
- 18C15 Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples
- 18C20 Algebras and Kleisli categories associated with monads
- 18C30 Sketches and generalizations
- 18C35 Accessible and locally presentable categories
- 18C50 Categorical semantics of formal languages
- 18C99 None of the above, but in this section

**▼****18Dxx Categories with structure**- 18D05 Double categories,
*2*-categories, bicategories and generalizations - 18D10 Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories
- 18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
- 18D20 Enriched categories (over closed or monoidal categories)
- 18D25 Strong functors, strong adjunctions
- 18D30 Fibered categories
- 18D35 Structured objects in a category (group objects, etc.)
- 18D50 Operads
- 18D99 None of the above, but in this section

- 18D05 Double categories,
**▼****18Exx Abelian categories**- 18E05 Preadditive, additive categories
- 18E10 Exact categories, abelian categories
- 18E15 Grothendieck categories
- 18E20 Embedding theorems
- 18E25 Derived functors and satellites
- 18E30 Derived categories, triangulated categories
- 18E35 Localization of categories
- 18E40 Torsion theories, radicals
- 18E99 None of the above, but in this section

**►****18Fxx Categories and geometry****►****18Gxx Homological algebra**

**▼****19-XX***K*-theory**►****19-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****►****19Axx Grothendieck groups and***K*_{0}**►****19Bxx Whitehead groups and***K*_{1}**►****19Cxx Steinberg groups and***K*_{2}**►****19Dxx Higher algebraic***K*-theory**►****19Exx***K*-theory in geometry**►****19Fxx***K*-theory in number theory**►****19Gxx***K*-theory of forms**▼****19Jxx Obstructions from topology****►****19Kxx***K*-theory and operator algebras**►****19Lxx Topological***K*-theory- 19M05 Miscellaneous applications of
*K*-theory

**▼****20-XX Group theory and generalizations****►****20-00 General reference works (handbooks, dictionaries, bibliographies, etc.)****►****20Axx Foundations****▼****20Bxx Permutation groups**- 20B05 General theory for finite groups
- 20B07 General theory for infinite groups
- 20B10 Characterization theorems
- 20B15 Primitive groups
- 20B20 Multiply transitive finite groups
- 20B22 Multiply transitive infinite groups
- 20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
- 20B27 Infinite automorphism groups
- 20B30 Symmetric groups
- 20B35 Subgroups of symmetric groups
- 20B40 Computational methods
- 20B99 None of the above, but in this section

**▼****20Cxx Representation theory of groups**- 20C05 Group rings of finite groups and their modules
- 20C07 Group rings of infinite groups and their modules
- 20C08 Hecke algebras and their representations
- 20C10 Integral representations of finite groups
- 20C11
*p*-adic representations of finite groups - 20C12 Integral representations of infinite groups
- 20C15 Ordinary representations and characters
- 20C20 Modular representations and characters
- 20C25 Projective representations and multipliers
- 20C30 Representations of finite symmetric groups
- 20C32 Representations of infinite symmetric groups
- 20C33 Representations of finite groups of Lie type
- 20C34 Representations of sporadic groups
- 20C35 Applications of group representations to physics
- 20C40 Computational methods
- 20C99 None of the above, but in this section

**►****20Dxx Abstract finite groups****►****20Exx Structure and classification of infinite or finite groups****►****20Fxx Special aspects of infinite or finite groups****▼****20Gxx Linear algebraic groups (classical groups)**- 20G05 Representation theory
- 20G10 Cohomology theory
- 20G15 Linear algebraic groups over arbitrary fields
- 20G20 Linear algebraic groups over the reals, the complexes, the quaternions
- 20G25 Linear algebraic groups over local fields and their integers
- 20G30 Linear algebraic groups over global fields and their integers
- 20G35 Linear algebraic groups over adèles and other rings and schemes
- 20G40 Linear algebraic groups over finite fields
- 20G42 Quantum groups (quantized function algebras) and their representations
- 20G45 Applications to physics
- 20G99 None of the above, but in this section

**►****20Hxx Other groups of matrices****►****20Jxx Connections with homological algebra and category theory****►****20Kxx Abelian groups**- 20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
**▼****20Mxx Semigroups**- 20M05 Free semigroups, generators and relations, word problems
- 20M07 Varieties of semigroups
- 20M10 General structure theory
- 20M11 Radical theory
- 20M12 Ideal theory
- 20M14 Commutative semigroups
- 20M15 Mappings of semigroups
- 20M17 Regular semigroups
- 20M18 Inverse semigroups
- 20M19 Orthodox semigroups
- 20M20 Semigroups of transformations, etc.
- 20M25 Semigroup rings, multiplicative semigroups of rings
- 20M30 Representation of semigroups; actions of semigroups on sets
- 20M35 Semigroups in automata theory, linguistics, etc.
- 20M50 Connections of semigroups with homological algebra and category theory
- 20M99 None of the above, but in this section

**►****20Nxx Other generalizations of groups**- 20P05 Probabilistic methods in group theory

**▼****Analysis****►****22-XX Topological groups, Lie groups****►****26-XX Real functions****►****28-00 Measure and integration****►****30-XX Functions of a complex variable****►****31-XX Potential theory****►****32-XX Several complex variables and analytic spaces****►****33-XX Special functions (EDD 000 deals with the properties of functions as functions)**

**►****Differential Equations****►****Transformations****►****Geometry****►****Statistics****►****Applied Mathematics****►****Didactics**

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