# User:Lethe/list of categories

In category theory, categories are the main object of study. The following is a list of important categories, and a glossary of named categories.

## Table of categories

Legend
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
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.

## Categories

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
Aff or AffSch affine schemes
K-Alg or AlgK associative unital algebras over field K homomorphisms y y yy 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
${\displaystyle {\mathfrak {Dif}}}$ 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 ${\displaystyle {}_{R}{\mathfrak {M}}}$ left R-modules R-linear homomorphisms y Ab y n
Mod-S or ${\displaystyle {\mathfrak {M}}_{S}}$ right S-modules S-linear homomophisms y Ab y n
R-Mod-S or ${\displaystyle {}_{R}{\mathfrak {M}}_{S}}$ 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
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

## what are the abbreviated names for these categories? I could take a guess

• category of presheaves ( a functor category) and the category of sheaves Sh(X) and Psc(X)
• category of CW complexes
• category of complex manifolds
• measure spaces
• Cauchy spaces
• Riemannian manifolds
• projective spaces over K
• Affine spaces over K
• stein manifolds
• category of structures for a given language
• varieties (affine, quasi-affine, projective, quasi-projective). dually equivalent to fgDom

## to be inserted

• Tych
• Kähler
• Rel category of sets and relations between them (not concrete)
• comUnC*Alg commutative unital C*-algebras with unital *-homomorphisms
• compTopGrp compact topological groups
• fdVectK finite dimensional vector spaces
• CoAlgK coalgebras
• BiAlgK bialgebras
• FRL Fröhlicher spaces (ccc) a full subcategory of DSP
• Frm
• G-Set of group actions. a topos
• TVS
• LCTVS
• CPO complete partial orders
• DCPO
• CABA complete atomic boolean algebras. dually equivalent to Set by some version of Stone
• Prof the category of categories, profunctors, and natural tranformations
• GRPO G-relative pushouts and RPO relative pushouts
• Bun bunch contexts
• Algτ algebras (in the sense of universal algebra) with signature τ.
• Chu(V,k) Chu spaces over V valued in k

### classes of categories

• any preordered set is a category with elements for objects and "<" as the morphisms. It follows that any ordinal is a category.
• any monoid is a category with one object and elements as morphisms
• consequently any group is a category with one object with elements as morphisms and has the categorical property that all its morphisms are isomorphisms
• a category is generated by any graph
• given any set, the discrete category on that set has the elements as objects and only identity morphisms
• given any category C, we may form the dual category Cop
• given two categories C and D, we may form the product category CxD
• given two categories C and D, we may form the functor category DC
• given a category C and an object b of C, the comma category (bC) of objects under b is arrows from b. The comma category (Cb) of objects over b is arrows to b. More generally, given two functors F and G to C, one may form the comma category (FG)
• assuming the axiom of choice, every category has a skeleton, whose objects are representatives of the isomorphism classes of the category.