18-XX
|
Category theory; abstract homological algebra
{See
13Dxx
for commutative rings,
16Exx
for associative rings,
20Jxx
for groups,
57Txx
for topological groups and related structures; 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 especially
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.)
|
18A32
|
Factorization of morphisms, substructures, quotient structures, congruences, amalgams
|
18A35
|
Categories admitting limits (complete categories), functors preserving limits, completions
|
18A40
|
Adjoint functors (representable functors, universal constructions, reflective subcategories, reflections, etc.), constructions of adjoints (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]
|
18B20
|
Categories of machines, automata, operative categories
[See also
03D05,
68Qxx]
|
18B25
|
Topoi
[See also
03G30]
|
18B30
|
Categories of topological spaces and continuous mappings
[See also
54-XX]
|
18B35
|
Preorders, orders and lattices (viewed as categories)
[See also
06-XX]
|
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]
|
18C10
|
Theories (e.g.
Algebraic theories), structure, and semantics
[See also
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]
|
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]
|
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
[See also
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]
|
18E25
|
Derived functors and satellites
|
18E30
|
Derived categories, triangulated categories
|
18E35
|
Localization of categories
|
18E40
|
Torsion theories, radicals
[See also
13D30,
16S90]
|
18E99
|
None of the above, but in this section
|
18Fxx
|
Categories and geometry
|
18F05
|
Local categories and functors
|
18F10
|
Grothendieck topologies
[See also
14F20]
|
18F15
|
Abstract manifolds and fiber bundles
[See also
55Rxx,
57Pxx]
|
18F20
|
Presheaves and sheaves
[See also
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]
|
18F30
|
Grothendieck groups
[See also
13D15,
16E20,
19Axx]
|
18F99
|
None of the above, but in this section
|
18Gxx
|
Abstract homological algebra
[See also
13Dxx,
16Exx,
20Jxx,
55Nxx,
55Uxx,
57Txx]
|
18G05
|
Projectives and injectives
[See also
13C10,
13C11,
16D40,
16D50]
|
18G10
|
Resolutions; derived functors
[See also
13D02,
16E05,
18E25]
|
18G15
|
Ext and Tor, generalizations, K\"unneth 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]
|
18G50
|
Nonabelian homological algebra
|
18G55
|
Nonabelian homotopical algebra
|
18G60
|
Other (co)homology theories
[See also
19D55,
46L80,
58J20,
58J22]
|
18G99
|
None of the above, but in this section
|