Comparison of categories

Selected: category of commutative monoids, category of commutative rings. Choose different categories.

Property	$\CMon$	$\CRing$
abelian
accessible
additive
balanced
Barr-coexact
Barr-exact
binary copowers
binary coproducts
binary powers
binary products
biproducts
cartesian closed
cartesian filtered colimits
Cauchy complete
CIP
co-Malcev
coaccessible
cocartesian coclosed
cocartesian cofiltered limits
cocomplete
codistributive
coequalizers
coequalizers of kernel pairs
coextensive
cofiltered
cofiltered limits
cofiltered-limit-stable epimorphisms
cogenerating set
cogenerator
cokernel pairs
cokernels
complete
connected
connected colimits
connected limits
conormal
copowers
coproducts
coquotients of cocongruences
core-connected
core-thin
coreflexive equalizers
coregular
cosifted
cosifted limits
counital
countable
countable copowers
countable coproducts
countable powers
countable products
countably codistributive
countably coextensive
countably distributive
countably extensive
CSP
direct
directed colimits
directed limits
discrete
disjoint coproducts
disjoint finite coproducts
disjoint finite products
disjoint products
distributive
effective cocongruences
effective congruences
elementary topos
epi-regular
equalizers
equalizers of cokernel pairs
essentially countable
essentially discrete
essentially finite
essentially small
exact cofiltered limits
exact filtered colimits
extensive
extremal cogenerating set
extremal cogenerator
extremal generating set
extremal generator
filtered
filtered colimits
filtered-colimit-stable monomorphisms
finitary algebraic
finite
finite copowers
finite coproducts
finite powers
finite products
finitely accessible
finitely cocomplete
finitely complete
gaunt
generalized variety
generating set
generator
Grothendieck abelian
Grothendieck topos
groupoid
infinitary codistributive
infinitary coextensive
infinitary distributive
infinitary extensive
inhabited
initial object
inverse
kernel pairs
kernels
left cancellative
locally cartesian closed
locally cocartesian coclosed
locally copresentable
locally essentially small
locally finite
locally finitely multi-presentable
locally finitely presentable
locally multi-presentable
locally poly-presentable
locally presentable
locally small
locally ℵ₁-presentable
Malcev
mono-regular
multi-algebraic
multi-cocomplete
multi-complete
multi-initial object
multi-terminal object
natural numbers object
normal
one-sorted finitary algebraic
one-way
parametrized natural numbers object
pointed
powers
preadditive
pretopos
products
pullbacks
pushouts
quasitopos
quotient object classifier
quotient-trivial
quotients of congruences
reflexive coequalizers
regular
regular quotient object classifier
regular subobject classifier
regular-quotient-trivial
regular-subobject-trivial
right cancellative
self-dual
semi-strongly connected
sequential colimits
sequential limits
sifted
sifted colimits
skeletal
small
split abelian
strict initial object
strict terminal object
strongly connected
subobject classifier
subobject-trivial
terminal object
thin
trivial
unital
well-copowered
well-powered
wide pullbacks
wide pushouts
zero morphisms
ℵ₁-accessible
ℵ₁-cofiltered
ℵ₁-cofiltered limits
ℵ₁-filtered
ℵ₁-filtered colimits
ℵ₂-small copowers
ℵ₂-small coproducts
ℵ₂-small powers
ℵ₂-small products