Properties of categories
Found 188 properties (120 grouped)
- is abelian
- is accessible / coaccessible
- is additive
- is balanced
- is Barr-coexact / Barr-exact
- has binary copowers / binary powers
- has binary coproducts / binary products
- has biproducts
- is cartesian closed / cocartesian coclosed
- has cartesian filtered colimits / cocartesian cofiltered limits
- is Cauchy complete
- satisfies CIP / CSP
- is Malcev / co-Malcev
- is complete / cocomplete
- is distributive / codistributive
- has equalizers / coequalizers
- has equalizers of cokernel pairs / coequalizers of kernel pairs
- is extensive / coextensive
- is filtered / cofiltered
- has filtered colimits / cofiltered limits
- has filtered-colimit-stable monomorphisms / cofiltered-limit-stable epimorphisms
- has a generating set / cogenerating set
- has a generator / cogenerator
- has kernel pairs / cokernel pairs
- has kernels / cokernels
- is concretizable
- is connected
- has connected limits / connected colimits
- is normal / conormal
- has powers / copowers
- has products / coproducts
- has quotients of congruences / coquotients of cocongruences
- is core-connected
- is core-thin
- has reflexive coequalizers / coreflexive equalizers
- is regular / coregular
- is sifted / cosifted
- has sifted colimits / cosifted limits
- is total / cototal
- is unital / counital
- is countable
- has countable powers / countable copowers
- has countable products / countable coproducts
- is countably distributive / countably codistributive
- is countably extensive / countably coextensive
- is direct / inverse
- has directed colimits / directed limits
- is discrete
- has disjoint coproducts / disjoint products
- has disjoint finite coproducts / disjoint finite products
- has effective cocongruences / effective congruences
- is an elementary topos
- is epi-regular / mono-regular
- is essentially countable
- is essentially discrete
- is essentially finite
- is essentially small
- has exact cofiltered limits / exact filtered colimits
- has an extremal cogenerating set / extremal generating set
- has an extremal cogenerator / extremal generator
- is finitary algebraic
- is finite
- has finite copowers / finite powers
- has finite coproducts / finite products
- is finitely accessible
- is finitely cocomplete / finitely complete
- is gaunt
- is a generalized variety
- is Grothendieck abelian
- is a Grothendieck topos
- is a groupoid
- is infinitary codistributive / infinitary distributive
- is infinitary coextensive / infinitary extensive
- is inhabited
- has an initial object / terminal object
- is left cancellative / right cancellative
- is locally cartesian closed / locally cocartesian coclosed
- is locally copresentable / locally presentable
- is locally essentially small
- is locally finite
- is locally finitely multi-presentable
- is locally finitely presentable
- is locally multi-presentable
- is locally poly-presentable
- is locally small
- is locally ℵ₁-presentable
- is multi-algebraic
- is multi-cocomplete / multi-complete
- has a multi-initial object / multi-terminal object
- has a natural numbers object
- is one-sorted finitary algebraic
- is one-way
- has a parametrized natural numbers object
- is pointed
- is preadditive
- is a pretopos
- has pullbacks / pushouts
- is a quasitopos
- has a quotient object classifier / subobject classifier
- is quotient-trivial / subobject-trivial
- has a regular quotient object classifier / regular subobject classifier
- is regular-quotient-trivial / regular-subobject-trivial
- is self-dual
- is semi-strongly connected
- has sequential colimits / sequential limits
- is skeletal
- is small
- is split abelian
- has a strict initial object / strict terminal object
- is strongly connected
- is thin
- is trivial
- is well-copowered / well-powered
- has wide pullbacks / wide pushouts
- has zero morphisms
- is ℵ₁-accessible
- is ℵ₁-cofiltered / ℵ₁-filtered
- has ℵ₁-cofiltered limits / ℵ₁-filtered colimits
- has ℵ₂-small copowers / ℵ₂-small powers
- has ℵ₂-small coproducts / ℵ₂-small products