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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****▼****18Bxx Special categories**- 18B05 Category of sets, characterizations
- 18B10 Category of relations, additive relations
- 18B15 Embedding theorems, universal categories
- 18B20 Categories of machines, automata, operative categories
- 18B25 Topoi
- 18B30 Categories of topological spaces and continuous mappings
- 18B35 Preorders, orders and lattices (viewed as categories)
- 18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
- 18B99 None of the above, but in this section

**▼****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****►****18Fxx Categories and geometry****▼****18Gxx Homological algebra**- 18G05 Projectives and injectives
- 18G10 Resolutions; derived functors
- 18G15 Ext and Tor, generalizations, Künneth formula
- 18G20 Homological dimension
- 18G25 Relative homological algebra, projective classes
- 18G30 Simplicial sets, simplicial objects (in a category)
- 18G35 Chain complexes
- 18G40 Spectral sequences, hypercohomology
- 18G50 Nonabelian homological algebra
- 18G55 Homotopical algebra
- 18G60 Other (co)homology theories
- 18G99 None of the above, but in this section

**▼****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****►****20Cxx Representation theory of groups****►****20Dxx Abstract finite groups****►****20Exx Structure and classification of infinite or finite groups****▼****20Fxx Special aspects of infinite or finite groups**- 20F05 Generators, relations, and presentations
- 20F06 Cancellation theory; application of van Kampen diagrams
- 20F10 Word problems, other decision problems, connections with logic and automata
- 20F12 Commutator calculus
- 20F14 Derived series, central series, and generalizations
- 20F16 Solvable groups, supersolvable groups
- 20F17 Formations of groups, Fitting classes
- 20F18 Nilpotent groups
- 20F19 Generalizations of solvable and nilpotent groups
- 20F22 Other classes of groups defined by subgroup chains
- 20F24 FC-groups and their generalizations
- 20F28 Automorphism groups of groups
- 20F29 Representations of groups as automorphism groups of algebraic systems
- 20F34 Fundamental groups and their automorphisms
- 20F36 Braid groups; Artin groups
- 20F38 Other groups related to topology or analysis
- 20F40 Associated Lie structures
- 20F45 Engel conditions
- 20F50 Periodic groups; locally finite groups
- 20F55 Reflection and Coxeter groups
- 20F60 Ordered groups
- 20F65 Geometric group theory
- 20F67 Hyperbolic groups and nonpositively curved groups
- 20F69 Asymptotic properties of groups
- 20F99 None of the above, but in this section

**►****20Gxx Linear algebraic groups (classical groups)****►****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****►****20Nxx Other generalizations of groups**- 20P05 Probabilistic methods in group theory

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

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