category of graded modules over a graded ring

Notation: $\grMod_G(R)$
Objects: $G$ -graded modules over a $G$ -graded ring $R$
Morphisms: $R$ -linear $G$ -graded maps
Children: $\Ch(\Ab)$ , $\Ab^{(\IN,\leq)}$ , $\grAb$
Related categories: $\FiltVect_K$ , $R{-}\Mod$
nLab Link

Here, $G$ is a monoid (written additively), $R = (R_n)_{n \in G}$ is a non-trivial $G$ -graded ring, and we consider $G$ -graded $R$ -modules, consisting of abelian groups $(M_n)_{n \in G}$ and bilinear maps $R_n \times M_m \to M_{n+m}$ satisfying the module axioms. Typical examples are $G = \IN$ and $G = \IZ$ , both with respect to addition.

Satisfied Properties

Assigned properties

Deduced properties

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 skeletal
does not satisfy CSP

Deduced properties*

is not discrete
is not gaunt
is not direct
does not have cofiltered-limit-stable epimorphisms
is not inverse
is not self-dual
is not trivial
is not a groupoid
does not have exact cofiltered limits
is not right cancellative
is not quotient-trivial
is not essentially finite
does not have a parametrized natural numbers object
is not thin
is not core-connected
is not essentially discrete
does not have a strict initial object
is not finite
does not have a regular subobject classifier
is not regular-subobject-trivial
is not regular-quotient-trivial
does not have a strict terminal object
does not have a regular quotient object classifier
is not left cancellative
is not cartesian closed
is not one-way
is not countably distributive
is not distributive
is not extensive
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

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Undecidable properties

There are 2 properties for which it cannot be decided if they are satisfied or not.

Special objects

terminal object: the zero graded module with $0_n \coloneqq 0$
initial object: the zero graded module with $0_n \coloneqq 0$
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