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 -homomorphismsn
Adj small categories adjunctions n
Aff or AffSch affine schemes
K -Alg or Alg K 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
n Cob (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 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
D
i
f
{\displaystyle {\mathfrak {Dif}}}
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 groupsy
pAb
DLat distributive lattices
Dom integral domains ring homomorphisms y
Ab
Dom m integral domains ring monomorphisms
y
Ab
Ens V 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
HopfAlg K Hopf algebras y
LCA locally compact abelian groups homomorphisms
y
z
pAb
y
n
dually isomorphic to itself by Pontryagin duality
Lconn locally connected spacescontinuous 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
R
M
{\displaystyle {}_{R}{\mathfrak {M}}}
left R -modules
R -linear homomorphismsy
Ab
y
n
Mod -S or
M
S
{\displaystyle {\mathfrak {M}}_{S}}
right S -modules
S -linear homomophismsy
Ab
y
n
R -Mod -S or
R
M
S
{\displaystyle {}_{R}{\mathfrak {M}}_{S}}
bimodules bilinear homomorphisms
y
Ab
y
n
Matr K 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 Vect K 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 -modulesy
Ab
y
n
Set 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 * 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 hTop topological spaces homotopy classes of mapsy
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 varietiesregular maps
y
f/f
dually equivalent to fgDom by an elementary result of algebraic geometry
VB K vector bundles with fibres over field K bundle morphisms y
y/y
n/n
add
y
n
Tangent bundle a functor from Smooth .
VB K X ) or Vect K 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 O X P (C ∞ (X )) for X compact by Swan's theorem .
Vect K 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
