In category theory, categories are the main object of study. The following is a list of important categories, and a glossary of named categories.
symbol | meaning | keys and comments |
---|---|---|
c | concrete category | objects and morphisms can be constructed as sets and functions |
/ | quotient objects | |
⊂ | subobjects | |
∏ | products | n: no objects have; c: some objects have; f: any finite number have; y: all objects; fbi:finite number have a biproduct; bi: all have biproduct |
∐ | coproducts | n: no objects have; c: some objects have; f: any finite number have; y: all objects |
= | equalizers | |
cq | coequalizers | |
i | initial object | |
t | terminal object | |
z | zero object | |
+ | additivity | |
→ | complete | |
← | cocomplete | |
⊗ | monoidal | |
ccc | Cartesian closed | |
y | yes. a category has a given property | |
a | all. For products or coproducts, all (small) collections of objects have the product or coproduct | |
f | finite. For products or coproducts, all finite collections of objects have product or coproduct. | |
bi | finite biproducts. All finite collections of objects in a pre-additive category have biproduct. |
Category | Objects | morphisms | c | //⊂ | ∏/ ∐ | =/ cq | i/ t/ z | + | →/ ← | ⊗ | ccc | comments |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Ab | abelian groups | group homomorphisms | y | y/y | y/ y | y/y | 0 | Ab | y/y | y | a full reflective subcategory of Grp. Isomorphic to Z-Mod. Abelianization a functor from Grp. | |
AbF | filtered abelian groups | y | y/y | 0 | pAb | |||||||
AbT | topological abelian groups | homomorphisms | y | y/y | y/y | y/y | 0 | pAb | ||||
Act-S | semiautomata | S-homomorphisms | n | |||||||||
Adj | small categories | adjunctions | n | |||||||||
Aff or AffSch | affine schemes | |||||||||||
K-Alg or AlgK | associative unital algebras over field K | homomorphisms | y | y | y y | 0 | Ab | y | y | n | ||
AlgSet/K | algebraic sets | regular maps | y | y/y | n | n | n | |||||
Bool | Boolean algebras | homomorphisms | y | n | y | dually equivalent to Stone by Stone's representation theorem | ||||||
CAb | compact abelian groups | group homomorphisms | y | Ab | y | n | ||||||
Cat | small categories | functors | n | y/n | t: 1 | n | y | y | y | with natural transformations, forms a 2-category | ||
CGHaus | compactly generated Hausdorff spaces | y | y/ y | n | n | y | y | used as a replacement for Top which has the benefit of being Cartesian closed | ||||
Cls | classes | functions | n1 | |||||||||
nCob | (n−1)-dimensional manifolds | n-dimensional cobordisms | n | n | n | n | ||||||
CohLoc | coherent locales | equivalent to CohSp by Stone duality | ||||||||||
CohSp | coherent sober spaces | equivalent to CohLoc by Stone duality | ||||||||||
Comp | chain complexes | y | 0 | Ab | ||||||||
CompBool | complete Boolean algebras | homomorphisms | y | n | y | |||||||
CompHaus or HComp | compact Hausdorff spaces | continuous maps | y | n | n | y | dually equivalent to comUnC*Alg by Gelfand representation. A full reflective subcategory of completely regular Hausdorff spaces by Stone-Čech compactification. | |||||
Compmet | complete metric spaces | y | n | n | y | |||||||
comUnC*Alg | commutative unital C* algebras | *-homomorphisms | y | |||||||||
CRng | commutative rings | ring homomorphisms | y | y/y | y/ y | y/y | 0 | Ab | y | dually equivalent to AffSch | ||
DGA | differential graded algebras | |||||||||||
or DSP | diffeological or differential spaces | y | y/y | y/y | y/y | t:z | n | y/y | n | a replacement for Diff which has the benefit of being complete and cocomplete (but is not Cartesian closed) | ||
Diff or Smooth or Sm | smooth manifolds | smooth maps | y | y/y | n/n | n | n/n | n | ||||
Div | divisible abelian groups | y | pAb | |||||||||
DLat | distributive lattices | |||||||||||
Dom | integral domains | ring homomorphisms | y | Ab | ||||||||
Domm | integral domains | ring monomorphisms | y | Ab | ||||||||
EnsV | subsets of universal set V | functions | y | y/ y | y/ y | t | n | y | y | |||
Euclid | Euclidean vector spaces | orthogonal transformations | y | 0 | Ab | |||||||
Fin | equivalence class of finite sets | functions | y | y | y | y | t | n | y | the skeletal category of FinSet. Isomorphic to ω. | ||
FinOrd | finite ordinals | monotonic functions | y | n | n | y | ||||||
FinSet | finite sets | functions | y | n | n | y | ||||||
Fld | fields | field homomorphisms | y | y/n | n/n | n/n | n | n | n | n | n | all morphisms are monic |
Frm | frames | defined to be the opposite category of Loc | ||||||||||
Grp | groups | group homomorphisms | y | y/ y | y/ y | n/y | z | n | y | y | n | |
Grph | directed graphs | n | n | y/y | y | comma category (Set↓Δ) | ||||||
Ha | Heyting algebras | |||||||||||
Haus | Hausdorff spaces | continuous maps | y | y/y | n | n | y | n | ||||
Hilb | Hilbert spaces | linear maps | y | y | y | z | Ab | y | y | n | ||
HopfAlgK | Hopf algebras | y | ||||||||||
LCA | locally compact abelian groups | homomorphisms | y | z | pAb | y | n | dually isomorphic to itself by Pontryagin duality | ||||
Lconn | locally connected spaces | continuous maps | y | y/ y | n | n | ||||||
LieAlg | Lie algebras | Lie algebra homomorphisms | y | y/y | bi | 0 | Ab | functor from LieGrp | ||||
LieGrp | Lie groups | smooth homomorphisms | y | n | n | |||||||
Loc | locales | the object of study in pointless topology. See Stone duality | ||||||||||
Mag | magmas | homomorphisms | y | n | n | |||||||
Mod | modules | morphisms of modules and underlying rings | y | a fibered category over Rng | ||||||||
R-Mod or | left R-modules | R-linear homomorphisms | y | Ab | y | n | ||||||
Mod-S or | right S-modules | S-linear homomophisms | y | Ab | y | n | ||||||
R-Mod-S or | bimodules | bilinear homomorphisms | y | Ab | y | n | ||||||
MatrK | matrices over field (or sometimes ring) K | y | Ab | y | n | |||||||
Med | medial magmas | homomorphisms | y | n | y | n | ||||||
Met | metric spaces | short maps | y | n | y | n | ||||||
Mon | monoids | monoid homomorphisms | y | n | y | n | ||||||
MonCat | monoidal categories | strict morphisms | y | n | y | n | ||||||
Ord | preordered sets | monotonic functions | y | c/ c | n | y | n | |||||
P(R) | finitely generated projective modules over R | |||||||||||
Rel | sets | binary relations | n | |||||||||
Rep(G) | K-linear representations of G | functor category from G to VectK. Isomorphic to KG-Mod. | ||||||||||
Rng | rings | ring homomorphisms | y | i: Z | Ab | y | n | |||||
Sch | schemes | rational maps | y | t: Spec(Z) | n | y | n | |||||
Ses-A | short exact sequences of A-modules | y | Ab | y | n | |||||||
Set or Sets | sets | functions | y | y/ y | y/y | t: * | n | y | y | |||
Set* | pointed sets | basepoint preserving functions | y | n | y | n | comma category (*↓Set) | |||||
SFrm | frames | dually equivalent to Sob by Stone duality | ||||||||||
SLoc | spatial locales | opposite category of SFrm, thus equivalent to Sob by Stone duality | ||||||||||
Smgrp | semigroups | homomorphisms | y | n | y | n | ||||||
Sob | sober spaces | dually equivalent to SFrm by Stone duality | ||||||||||
Stone | Stone spaces | dually equivalent to Bool by Stone's representation theorem | ||||||||||
StrAlgSet/K | structured algebraic sets | y | n | y | n | |||||||
Top | topological spaces | continuous maps | y | y/ y | y/ y | y/y | t:* | n | y/y | n | ||
Top* or Top• | pointed topological spaces | basepoint preserving continuous maps | y | y/y | y/ y | z:* | n | y | n | comma category (*↓Top). fundamental group is a functor to Grp. | ||
Toph or hTop | topological spaces | homotopy classes of maps | y | n | y | n | ||||||
TOP(X) or O(X) or Open(X) | open sets in the topological space X | inclusions | n | y/ y | i: ∅ t:X | n | y | y | ||||
Uni | uniform spaces | uniformly continuous functions | y | n | n | n | ||||||
US | unary systems | [1] | ||||||||||
US1 | pointed unary systems | i:N | the natural numbers are initial. More generally, a natural number object is initial in pointed unary systems over some category | |||||||||
varieties | affine,quasi,projective,quasi-projective varieties | regular maps | y | f/f | dually equivalent to fgDom by an elementary result of algebraic geometry | |||||||
VBK | vector bundles with fibres over field K | bundle morphisms | y | y/y | n/n | add | y | n | Tangent bundle a functor from Smooth. | |||
VBK(X) or VectK(X) | vector bundles over X with fibres over field K | bundle morphisms | y | y/y | bi | n/n | 0 | add | y | n | equivalent to the category of locally free f.g. sheaves of OX-modules. Smooth vector bundles equivalent to P(C∞(X)) for X compact by Swan's theorem. | |
VectK or K-Vect | vector spaces over the field K | K- linear maps | y | Ab | y | n | ||||||
Vect(K,Z/2Z) | Z2- graded vector spaces | Z2-graded K-linear maps | y | Ab | y | n | ||||||
0 | the empty category | y | n | n | ||||||||
1 | one object | identity morphism | z:0 | n | n | n | ||||||
2 | i:0;t:1 | n | n | n | the ordinal 2 | |||||||
3 | i:0;t:2 | n | n | n | the ordinal 3 | |||||||
ω | y | y/ y | i:0 | n | the ordinal ω | |||||||
↓↓ | n | n | n | n |
In category theory, categories are the main object of study. The following is a list of important categories, and a glossary of named categories.
symbol | meaning | keys and comments |
---|---|---|
c | concrete category | objects and morphisms can be constructed as sets and functions |
/ | quotient objects | |
⊂ | subobjects | |
∏ | products | n: no objects have; c: some objects have; f: any finite number have; y: all objects; fbi:finite number have a biproduct; bi: all have biproduct |
∐ | coproducts | n: no objects have; c: some objects have; f: any finite number have; y: all objects |
= | equalizers | |
cq | coequalizers | |
i | initial object | |
t | terminal object | |
z | zero object | |
+ | additivity | |
→ | complete | |
← | cocomplete | |
⊗ | monoidal | |
ccc | Cartesian closed | |
y | yes. a category has a given property | |
a | all. For products or coproducts, all (small) collections of objects have the product or coproduct | |
f | finite. For products or coproducts, all finite collections of objects have product or coproduct. | |
bi | finite biproducts. All finite collections of objects in a pre-additive category have biproduct. |
Category | Objects | morphisms | c | //⊂ | ∏/ ∐ | =/ cq | i/ t/ z | + | →/ ← | ⊗ | ccc | comments |
---|---|---|---|---|---|---|---|---|---|---|---|---|
Ab | abelian groups | group homomorphisms | y | y/y | y/ y | y/y | 0 | Ab | y/y | y | a full reflective subcategory of Grp. Isomorphic to Z-Mod. Abelianization a functor from Grp. | |
AbF | filtered abelian groups | y | y/y | 0 | pAb | |||||||
AbT | topological abelian groups | homomorphisms | y | y/y | y/y | y/y | 0 | pAb | ||||
Act-S | semiautomata | S-homomorphisms | n | |||||||||
Adj | small categories | adjunctions | n | |||||||||
Aff or AffSch | affine schemes | |||||||||||
K-Alg or AlgK | associative unital algebras over field K | homomorphisms | y | y | y y | 0 | Ab | y | y | n | ||
AlgSet/K | algebraic sets | regular maps | y | y/y | n | n | n | |||||
Bool | Boolean algebras | homomorphisms | y | n | y | dually equivalent to Stone by Stone's representation theorem | ||||||
CAb | compact abelian groups | group homomorphisms | y | Ab | y | n | ||||||
Cat | small categories | functors | n | y/n | t: 1 | n | y | y | y | with natural transformations, forms a 2-category | ||
CGHaus | compactly generated Hausdorff spaces | y | y/ y | n | n | y | y | used as a replacement for Top which has the benefit of being Cartesian closed | ||||
Cls | classes | functions | n1 | |||||||||
nCob | (n−1)-dimensional manifolds | n-dimensional cobordisms | n | n | n | n | ||||||
CohLoc | coherent locales | equivalent to CohSp by Stone duality | ||||||||||
CohSp | coherent sober spaces | equivalent to CohLoc by Stone duality | ||||||||||
Comp | chain complexes | y | 0 | Ab | ||||||||
CompBool | complete Boolean algebras | homomorphisms | y | n | y | |||||||
CompHaus or HComp | compact Hausdorff spaces | continuous maps | y | n | n | y | dually equivalent to comUnC*Alg by Gelfand representation. A full reflective subcategory of completely regular Hausdorff spaces by Stone-Čech compactification. | |||||
Compmet | complete metric spaces | y | n | n | y | |||||||
comUnC*Alg | commutative unital C* algebras | *-homomorphisms | y | |||||||||
CRng | commutative rings | ring homomorphisms | y | y/y | y/ y | y/y | 0 | Ab | y | dually equivalent to AffSch | ||
DGA | differential graded algebras | |||||||||||
or DSP | diffeological or differential spaces | y | y/y | y/y | y/y | t:z | n | y/y | n | a replacement for Diff which has the benefit of being complete and cocomplete (but is not Cartesian closed) | ||
Diff or Smooth or Sm | smooth manifolds | smooth maps | y | y/y | n/n | n | n/n | n | ||||
Div | divisible abelian groups | y | pAb | |||||||||
DLat | distributive lattices | |||||||||||
Dom | integral domains | ring homomorphisms | y | Ab | ||||||||
Domm | integral domains | ring monomorphisms | y | Ab | ||||||||
EnsV | subsets of universal set V | functions | y | y/ y | y/ y | t | n | y | y | |||
Euclid | Euclidean vector spaces | orthogonal transformations | y | 0 | Ab | |||||||
Fin | equivalence class of finite sets | functions | y | y | y | y | t | n | y | the skeletal category of FinSet. Isomorphic to ω. | ||
FinOrd | finite ordinals | monotonic functions | y | n | n | y | ||||||
FinSet | finite sets | functions | y | n | n | y | ||||||
Fld | fields | field homomorphisms | y | y/n | n/n | n/n | n | n | n | n | n | all morphisms are monic |
Frm | frames | defined to be the opposite category of Loc | ||||||||||
Grp | groups | group homomorphisms | y | y/ y | y/ y | n/y | z | n | y | y | n | |
Grph | directed graphs | n | n | y/y | y | comma category (Set↓Δ) | ||||||
Ha | Heyting algebras | |||||||||||
Haus | Hausdorff spaces | continuous maps | y | y/y | n | n | y | n | ||||
Hilb | Hilbert spaces | linear maps | y | y | y | z | Ab | y | y | n | ||
HopfAlgK | Hopf algebras | y | ||||||||||
LCA | locally compact abelian groups | homomorphisms | y | z | pAb | y | n | dually isomorphic to itself by Pontryagin duality | ||||
Lconn | locally connected spaces | continuous maps | y | y/ y | n | n | ||||||
LieAlg | Lie algebras | Lie algebra homomorphisms | y | y/y | bi | 0 | Ab | functor from LieGrp | ||||
LieGrp | Lie groups | smooth homomorphisms | y | n | n | |||||||
Loc | locales | the object of study in pointless topology. See Stone duality | ||||||||||
Mag | magmas | homomorphisms | y | n | n | |||||||
Mod | modules | morphisms of modules and underlying rings | y | a fibered category over Rng | ||||||||
R-Mod or | left R-modules | R-linear homomorphisms | y | Ab | y | n | ||||||
Mod-S or | right S-modules | S-linear homomophisms | y | Ab | y | n | ||||||
R-Mod-S or | bimodules | bilinear homomorphisms | y | Ab | y | n | ||||||
MatrK | matrices over field (or sometimes ring) K | y | Ab | y | n | |||||||
Med | medial magmas | homomorphisms | y | n | y | n | ||||||
Met | metric spaces | short maps | y | n | y | n | ||||||
Mon | monoids | monoid homomorphisms | y | n | y | n | ||||||
MonCat | monoidal categories | strict morphisms | y | n | y | n | ||||||
Ord | preordered sets | monotonic functions | y | c/ c | n | y | n | |||||
P(R) | finitely generated projective modules over R | |||||||||||
Rel | sets | binary relations | n | |||||||||
Rep(G) | K-linear representations of G | functor category from G to VectK. Isomorphic to KG-Mod. | ||||||||||
Rng | rings | ring homomorphisms | y | i: Z | Ab | y | n | |||||
Sch | schemes | rational maps | y | t: Spec(Z) | n | y | n | |||||
Ses-A | short exact sequences of A-modules | y | Ab | y | n | |||||||
Set or Sets | sets | functions | y | y/ y | y/y | t: * | n | y | y | |||
Set* | pointed sets | basepoint preserving functions | y | n | y | n | comma category (*↓Set) | |||||
SFrm | frames | dually equivalent to Sob by Stone duality | ||||||||||
SLoc | spatial locales | opposite category of SFrm, thus equivalent to Sob by Stone duality | ||||||||||
Smgrp | semigroups | homomorphisms | y | n | y | n | ||||||
Sob | sober spaces | dually equivalent to SFrm by Stone duality | ||||||||||
Stone | Stone spaces | dually equivalent to Bool by Stone's representation theorem | ||||||||||
StrAlgSet/K | structured algebraic sets | y | n | y | n | |||||||
Top | topological spaces | continuous maps | y | y/ y | y/ y | y/y | t:* | n | y/y | n | ||
Top* or Top• | pointed topological spaces | basepoint preserving continuous maps | y | y/y | y/ y | z:* | n | y | n | comma category (*↓Top). fundamental group is a functor to Grp. | ||
Toph or hTop | topological spaces | homotopy classes of maps | y | n | y | n | ||||||
TOP(X) or O(X) or Open(X) | open sets in the topological space X | inclusions | n | y/ y | i: ∅ t:X | n | y | y | ||||
Uni | uniform spaces | uniformly continuous functions | y | n | n | n | ||||||
US | unary systems | [1] | ||||||||||
US1 | pointed unary systems | i:N | the natural numbers are initial. More generally, a natural number object is initial in pointed unary systems over some category | |||||||||
varieties | affine,quasi,projective,quasi-projective varieties | regular maps | y | f/f | dually equivalent to fgDom by an elementary result of algebraic geometry | |||||||
VBK | vector bundles with fibres over field K | bundle morphisms | y | y/y | n/n | add | y | n | Tangent bundle a functor from Smooth. | |||
VBK(X) or VectK(X) | vector bundles over X with fibres over field K | bundle morphisms | y | y/y | bi | n/n | 0 | add | y | n | equivalent to the category of locally free f.g. sheaves of OX-modules. Smooth vector bundles equivalent to P(C∞(X)) for X compact by Swan's theorem. | |
VectK or K-Vect | vector spaces over the field K | K- linear maps | y | Ab | y | n | ||||||
Vect(K,Z/2Z) | Z2- graded vector spaces | Z2-graded K-linear maps | y | Ab | y | n | ||||||
0 | the empty category | y | n | n | ||||||||
1 | one object | identity morphism | z:0 | n | n | n | ||||||
2 | i:0;t:1 | n | n | n | the ordinal 2 | |||||||
3 | i:0;t:2 | n | n | n | the ordinal 3 | |||||||
ω | y | y/ y | i:0 | n | the ordinal ω | |||||||
↓↓ | n | n | n | n |