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

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.

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
Adj	small categories	adjunctions	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
DGA	differential graded algebras
${\mathfrak {Dif}}$ 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 ${}_{R}{\mathfrak {M}}$	left R-modules	R-linear homomorphisms	y					Ab	y		n
Mod-S or ${\mathfrak {M}}_{S}$	right S-modules	S-linear homomophisms	y					Ab	y		n
R-Mod-S or ${}_{R}{\mathfrak {M}}_{S}$	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
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

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
• fdVect_K finite dimensional vector spaces
• CoAlg_K coalgebras
• BiAlg_K 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
• Alg_τ algebras (in the sense of universal algebra) with signature τ.
• Chu(V,k) Chu spaces over V valued in k

specific categories

• Ab: abelian groups
• Adj: small categories and adjunctions between them
• K-Alg: algebras over field K
• AlgSet/K: algebraic sets with regular maps
• Bool: Boolean algebras and their homomorphisms
• CAb: compact topological abelian groups
• Cat: all small categories with functors
• CGHaus: compactly generated Hausdorff spaces
• nCob: (n−1)-dimensional manifolds with n-dimensional cobordisms
• Comp: chain complexes
• CompBool: complete Boolean algebras
• CompHaus: compact Hausdorff spaces
• Compmet: complete metric spaces
• CRng: commutative rings
• ${\mathfrak {Dif}}$: diffeological spaces
• Diff or Smooth: smooth manifolds with smooth maps
• Div: divisible abelian groups
• Dom: integral domains
• Dom_m: integral domains with monomorphisms
• Ens_V: sets and functions within a given universal set V
• Euclid: Euclidean vector spaces with orthogonal transformations
• Fin: The skeletal category of finite sets
• Finord: finite ordinals
• FinSet: finite sets with functions
• Fld: fields with field homomorphisms
• Grp: all groups with their group homomorphisms
• Grph: directed graphs
• Haus: Hausdorff spaces
• Hilb: Hilbert spaces with linear maps
• LCA: locally compact abelian groups with continuous homomorphisms
• Lconn: locally connected spaces
• LieGrp: Lie groups
• Mag: The category consisting of all magmas with their homomorphisms
• R-Mod: left R-modules
• Mod-S: right S-modules
• R-Mod-S: bimodules
• Matr_K: matrices over field (or sometimes ring) K
• Med: The category consisting of all medial magmas with their homomorphisms
• Met: all metric spaces with short maps
• Mon: monoids with monoid homomorphisms
• Moncat: monoidal categories and strict morphisms
• Ord: all preordered sets with monotonic functions
• Rng: rings
• Sch: schemes
• Ses-A: short exact sequences of A-modules
• Set: sets with functions
• Set_*: pointed sets with base preserving functions
• Smgrp: semigroups
• StrAlgSet/K: structured algebraic sets
• Top: all topological spaces with continuous functions
• Toph: topological spaces with homotopy classes
• TOP(X): open sets in the topological space X with inclusions
• Uni: all uniform spaces with uniformly continuous functions
• Vect_K or K-Vect: vector spaces over the field K (which is held fixed) with their K-linear maps
• Vect(K,Z/2Z): Z2-graded vector spaces with graded maps
• 0: The empty category
• 1: The one object category with one morphism
• 2: The two object category with one morphism not an identity between the distinct objects (this is the ordinal number 2 viewed as a category)
• 3: The three object with one morphisms between each distinct pair of objects (this is the ordinal number 3 viewed as a category)
• ↓↓: The two object category with two morphisms from one object to the other

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 C^op
• 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 D^C
• given a category C and an object b of C, the comma category (b ↓ C) of objects under b is arrows from b. The comma category (C ↓ b) of objects over b is arrows to b. More generally, given two functors F and G to C, one may form the comma category (F ↓ G)
• assuming the axiom of choice, every category has a skeleton, whose objects are representatives of the isomorphism classes of the category.