category of cochain complexes of abelian groups
- Notation:
- Objects: sequences of abelian groups such that
- Morphisms: commutative diagrams
- Parent:
- Related categories: , ,
- nLab Link
For every category with zero morphisms, we can construct the category of cochain complexes in . In this entry, is the category of abelian groups. The category is isomorphic to the category of -graded modules over the -graded ring , with the grading concentrated in degrees and .
Satisfied Properties
Assigned properties
—
Deduced properties
- is locally small
- is complete
- is cocomplete
- is abelian
- is finitary algebraic
- is locally finitely presentable
- is a generalized variety
- is additive
- has cokernels
- is conormal
- has kernels
- is normal
- is regular
- is multi-algebraic
- has connected limits
- is finitely complete
- has equalizers
- has products
- is multi-complete
- is locally essentially small
- is coregular
- has connected colimits
- is finitely cocomplete
- has coequalizers
- has coproducts
- is multi-cocomplete
- is Malcev
- is finitely accessible
- has exact filtered colimits
- is locally ℵ₁-presentable
- has finite products
- is preadditive
- has biproducts
- has sifted colimits
- is ℵ₁-accessible
- has filtered-colimit-stable monomorphisms
- is locally finitely multi-presentable
- has effective congruences
- has a multi-terminal object
- has coreflexive equalizers
- is Cauchy complete
- has zero morphisms
- is filtered
- is mono-regular
- has powers
- has ℵ₂-small products
- has wide pullbacks
- is co-Malcev
- has finite coproducts
- has a multi-initial object
- has reflexive coequalizers
- is cofiltered
- has cosifted limits
- is epi-regular
- has copowers
- has ℵ₂-small coproducts
- has wide pushouts
- is unital
- is locally presentable
- is accessible
- has ℵ₁-filtered colimits
- has filtered colimits
- has quotients of congruences
- is Barr-exact
- has cartesian filtered colimits
- is sifted
- is balanced
- has countable products
- has ℵ₂-small powers
- has binary products
- has a terminal object
- has finite powers
- has cofiltered limits
- has pullbacks
- is counital
- has coquotients of cocongruences
- has effective cocongruences
- is cosifted
- has countable coproducts
- has ℵ₂-small copowers
- has binary coproducts
- has an initial object
- has finite copowers
- has pushouts
- is pointed
- has an extremal generating set
- is well-powered
- is well-copowered
- is locally multi-presentable
- is locally poly-presentable
- is connected
- satisfies CIP
- is ℵ₁-filtered
- has directed colimits
- has sequential limits
- has countable powers
- has binary powers
- is Barr-coexact
- has cocartesian cofiltered limits
- is ℵ₁-cofiltered
- has directed limits
- has sequential colimits
- has ℵ₁-cofiltered limits
- has countable copowers
- has binary copowers
- has a natural numbers object
- is inhabited
- has a generating set
- is strongly connected
- is semi-strongly connected
- has disjoint finite products
- has a generator
- has an extremal generator
- has disjoint finite coproducts
- is Grothendieck abelian
- has disjoint coproducts
- has disjoint products
- has a cogenerator
- has a cogenerating set
- has an extremal cogenerator
- has an extremal cogenerating set
Unsatisfied Properties
Assigned properties
- is not split abelian
- is not one-sorted finitary algebraic
Deduced properties*
- is not skeletal
- does not satisfy CSP
- is not trivial
- is not discrete
- is not gaunt
- is not direct
- does not have cofiltered-limit-stable epimorphisms
- is not inverse
- is not self-dual
- does not have a parametrized natural numbers object
- is not thin
- is not core-connected
- is not essentially discrete
- is not a groupoid
- does not have a strict initial object
- does not have a regular subobject classifier
- is not regular-subobject-trivial
- is not regular-quotient-trivial
- does not have exact cofiltered limits
- is not right cancellative
- is not quotient-trivial
- is not essentially finite
- does not have a strict terminal object
- does not have a regular quotient object classifier
- is not cartesian closed
- is not one-way
- is not countably distributive
- is not left cancellative
- is not distributive
- is not extensive
- is not finite
- does not have a subobject classifier
- is not subobject-trivial
- is not core-thin
- is not locally finite
- is not essentially small
- is not essentially countable
- is not a quasitopos
- is not cocartesian coclosed
- is not codistributive
- is not coextensive
- does not have a quotient object classifier
- is not locally copresentable
- is not locally cartesian closed
- is not infinitary distributive
- is not countably extensive
- is not small
- is not countable
- is not an elementary topos
- is not a pretopos
- is not locally cocartesian coclosed
- is not countably codistributive
- is not countably coextensive
- is not infinitary extensive
- is not a Grothendieck topos
- is not coaccessible
- is not infinitary codistributive
- is not infinitary coextensive
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: the zero complex
- initial object: the zero complex
- products: degree-wise defined direct products
- coproducts: degree-wise defined direct sums
Special morphisms
- isomorphisms: morphisms that are bijective in each degree
- monomorphisms: morphisms that are injective in each degree
- epimorphisms: morphisms that are surjective in each degree
- regular monomorphisms: same as monomorphisms
- regular epimorphisms: same as epimorphisms
Indistinguishable categories
These categories in the database currently have exactly the same properties as the category of cochain complexes of abelian groups. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.