List of Categories (Math, Category Theory)

list-of-categories.md

Ab, of abelian groups and homomorphisms, 4
Abfg, of finitely generated abelian groups and homomorphisms, 26
Abtf, of torsion-free abelian groups and homomorphisms, 141
AdjT, of adjunctions inducing a monad T and maps of adjunctions, 164
Affine, of affine planes and bijections that preserve and reflect incidence, 36
Ban, of real Banach spaces and continuous linear maps, 26
BG, a group (or monoid) G regarded as a category, with one object and elements of the group G as endomorphisms, 5
Card, the discrete category of cardinals, 138
CAT, of locally small categories and functors, 20
Cat, of small categories and functors, 20
C/c, the slice of a category C over an object c, 8
c/C, the slice of a category C under an object c, 8
cgHaus, of compactly generated Hausdorff spaces and continuous maps, 130
cHaus, of compact Hausdorff spaces and continuous maps, 140
ChR, of chain complexes of R-modules and chain maps, 4
CJ, of functors J → C and natural transformations, 44
Cluster, of clusters and refinements, 16
CMonoid, of commutative monoids and homomorphisms, 119
Cop, the opposite of a category C, 9
CRing, of commutative unital rings and homomorphisms, 8
CT, the category of algebras for a monad T on a category C, 160
CT , the Kleisli category for a monad T on a category C, 162
C × D, the product of categories C and D, 19
∆, of finite non-empty ordinals and order-preserving maps, 18
∆+, of finite ordinals and order-preserving maps, 193
DirGraph, of directed graphs and homomorphisms, 4
RF, the category of elements for a set-valued functor F, 66
End, of sets equipped with an endomorphism and commuting functions, 69
Euclid∗, of pointed Euclidean spaces and pointed differentiable functions, 14
F ↓G, the comma category for a pair of functors F : D → C and G: E → C, 22
Field, of fields and homomorphisms, 4
FieldEF, of intermediate fields F ⊂ K ⊂ E and homomorphisms fixing F, 21
Fin, of finite sets and functions, 8
Fin∗, of pointed finite sets and basepoint-preserving functions, equivalently the opposite of Segal’s Γ, 15
Finiso, of finite sets and bijections, 8
FinMetric, of finite metric spaces and distance non-increasing functions, 16
Finmono, of finite sets and injections, 105
Γ, Segal’s category, equivalent to the opposite of Fin∗, 35
Graph, of graphs and homomorphisms, 4
GrModR, of Z-graded R-modules and graded homomorphisms, 14
Group, of groups and homomorphisms, 4
Groupepi, of groups and epimorphisms, 23
Groupiso, of groups and isomorphisms, 22
Groupoid, of groupoids and functors, 14
Haus, of Hausdorff spaces and continuous maps,
129
Htpy, of spaces and homotopy class of continuous
maps, 5
Htpy∗
, of based spaces and basepoint-preserving
homotopy class of based continuous maps, 5
HtpyCW, of CW complexes and homotopy classes
of continuous maps, 54
J
/
, the category obtained from J by freely
adjoining an initial object, 76
J
.
, the category obtained from J by freely
adjoining a terminal object, 76
Lattice, of lattices and lattice morphisms, 174
Man, of smooth manifolds and smooth functions, 4
Man∗, of pointed smooth manifolds and pointed
smooth functions, 15
MatR, of positive integers and matrices with
coefficients in the unital ring R, 4
Meas, of measurable spaces and measurable
functions, 4
Measure, of measure spaces and equivalence
classes of measurable functions, 5
ModelT, of models for a set T of sentences in a
first-order language and homomorphisms, 4
ModR, of left R-modules and homomorphisms, 4
RMod, of right R-modules and homomorphisms,
20
Monoid, of monoids and homomorphisms, 20
n, the ordinal n regarded as a category, 5
OG, of subgroups of a group G represented as left
G-sets of left cosets and G-equivariant maps,
21
ω, the ordinal ω regarded as a category, 5
O(X), the poset of open subsets of a space X under
inclusion, 18
(P, ≤), a poset (or preorder) regarded as a category,
with elements of P as objects and with a
morphism x → y if and only if x ≤ y, 5
PCluster, of persistent clusters and refinements, 16
Poset, of partially-ordered sets and
order-preserving functions, 4
Proj|
, of projective planes with a distinguished line
and bijections that preserve and reflect
incidence, 36
rDirGraph, of reflexive directed graphs and
homomorphisms, 177
Ring, of unital rings and homomorphisms, 4, 41
Rng, of non-unital rings and homomorphisms, 8,
41
SET, of “large” sets and functions, 59
Set, of sets and functions, 4
Set∗, of pointed sets and basepoint-preserving
functions, 4
Set∂
, of sets and partially-defined functions, 21
ShvX, of sheaves (of sets) on a space X and natural
transformations, 141
skC, the skeleton of the category C, 34
TGX, the translation groupoid for a G-set X, 34
Top, of spaces and continuous maps, 4
Top∗
, of pointed spaces and basepoint-preserving
continuous maps, 4
Toppc
∗
, of path-connected based spaces and based
continuous maps, 33
VectBG
k
, of G-representations and equivariant
linear maps, 192
Vectbasis
k
, of finite-dimensional k-vector spaces
with a chosen basis and linear maps, 30
Vectfd
k
, of finite-dimensional k-vector spaces and
linear maps, 30
Vectk, of k-vector spaces and linear maps, 4