| 18-XX | | Prev: 17 | Up: Top | Next: 19 | |
| Category theory; homological algebra [xref: 16B50, 46Mxx] | |
| {For commutative rings see 13Dxx, for associative rings 16Exx, for groups 20Jxx, for topological groups and related structures 57Txx; see also 55Nxx and 55Uxx for algebraic topology} |
18-00|
General reference works (handbooks, dictionaries, bibliographies, etc.)
| 18-01 |
Instructional exposition (textbooks, tutorial papers, etc.)
| 18-02 |
Research exposition (monographs, survey articles)
| 18-03 |
Historical (must also be assigned at least one classification number from Section 01)
| 18-04 |
Explicit machine computation and programs (not the theory of computation or programming)
| 18-06 |
Proceedings, conferences, collections, etc.
| |
18Axx|
General theory of categories and functors
| 18A05 |
Definitions, generalizations
| 18A10 |
Graphs, diagram schemes, precategories [See especially 20L05]
| 18A15 |
Foundations, relations to logic and deductive systems [See also 03-XX]
| 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.)
[xref: 08B25]
| 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
| 18B05 |
Category of sets, characterizations [See also 03-XX]
| 18B10 |
Category of relations, additive relations
| 18B15 |
Embedding theorems, universal categories [See also 18E20]
[xref: 18E20]
| 18B20 |
Categories of machines, automata, operative categories [See also 03D05, 68Qxx]
[xref: 68Q70]
| 18B25 |
Topoi [See also 03G30]
[xref: 03G30]
| 18B30 |
Categories of topological spaces and continuous mappings [See also 54-XX]
[xref: 54B30]
| 18B35 |
Preorders, orders and lattices (viewed as categories) [See also 06-XX]
[xref: 06B35]
| 18B40 |
Groupoids, semigroupoids, semigroups, groups (viewed as categories) [See also 20Axx, 20L05, 20Mxx]
| 18B99 |
None of the above, but in this section
| |
18Cxx|
Categories and theories
| 18C05 |
Equational categories [See also 03C05, 08C05]
[xref: 03C05, 03G30, 08C05]
| 18C10 |
Theories (e.g. algebraic theories), structure, and semantics [See also 03G30]
[xref: 03G30]
| 18C15 |
Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples [See also 18Gxx]
| 18C20 |
Algebras and Kleisli categories associated with monads
| 18C30 |
Sketches and generalizations
| 18C35 |
Accessible and locally presentable categories
| 18C50 |
Categorical semantics of formal languages [See also 68Q55, 68Q65]
[xref: 68Q55, 68Q65]
| 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 [See also 19D23]
[xref: 19D23]
| 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.)
[xref: 14L17]
| 18D50 |
Operads [See also 55P48]
[xref: 55P48]
| 18D99 |
None of the above, but in this section
| |
18Exx|
Abelian categories
| 18E05 |
Preadditive, additive categories
| 18E10 |
Exact categories, abelian categories
| 18E15 |
Grothendieck categories
| 18E20 |
Embedding theorems [See also 18B15]
[xref: 18B15]
| 18E25 |
Derived functors and satellites
[xref: 18G10]
| 18E30 |
Derived categories, triangulated categories
[xref: 18G35]
| 18E35 |
Localization of categories
| 18E40 |
Torsion theories, radicals [See also 13D30, 16S90]
[xref: 13D30, 16S90]
| 18E99 |
None of the above, but in this section
| |
18Fxx|
Categories and geometry
[xref: 46M20]
| 18F05 |
Local categories and functors
| 18F10 |
Grothendieck topologies [See also 14F20]
| 18F15 |
Abstract manifolds and fiber bundles [See also 55Rxx, 57Pxx]
[xref: 55Rxx, 57Pxx]
| 18F20 |
Presheaves and sheaves [See also 14F05, 32C35, 32L10, 54B40, 55N30]
[xref: 14F05, 32C35, 32L10, 54B40, 55N30]
| 18F25 |
Algebraic $K$-theory and $L$-theory [See also 11Exx, 11R70, 11S70, 12-XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67]
[xref: 46L80]
| 18F30 |
Grothendieck groups [See also 13D15, 16E20, 19Axx]
[xref: 13D15, 16E20, 19Axx]
| 18F99 |
None of the above, but in this section
| |
18Gxx|
Homological algebra [See also 13Dxx, 16Exx, 20Jxx, 55Nxx, 55Uxx, 57Txx]
[xref: 18C15, 55Uxx]
| 18G05 |
Projectives and injectives [See also 13C10, 13C11, 16D40, 16D50]
| 18G10 |
Resolutions; derived functors [See also 13D02, 16E05, 18E25]
| 18G15 |
Ext and Tor, generalizations, Künneth formula [See also 55U25]
| 18G20 |
Homological dimension [See also 13D05, 16E10]
| 18G25 |
Relative homological algebra, projective classes
| 18G30 |
Simplicial sets, simplicial objects (in a category) [See also 55U10]
| 18G35 |
Chain complexes [See also 18E30, 55U15]
| 18G40 |
Spectral sequences, hypercohomology [See also 55Txx]
[xref: 55Txx]
| 18G50 |
Nonabelian homological algebra
| 18G55 |
Homotopical algebra
[xref: 57T30]
| 18G60 |
Other (co)homology theories [See also 19D55, 46L80, 58J20, 58J22]
[xref: 19D55]
| 18G99 |
None of the above, but in this section
| |