category of left R-modules

This is the category of left modules over a ring RR. It is the prototype of an abelian category. The category of right modules is the same with the opposite ring RopR^{\mathrm{op}}, hence not listed here. The non-properties refer to the case that the ring is non-trivial, since for the trivial ring we get a trivial category which has all properties anyway. The category R−ModR{-}\mathbf{Mod} is split abelian iff RR is a semisimple ring, so usually it isn't the case, which is why we have negated this property here.

Properties

Properties from the database

Deduced properties

  • is locally essentially small
    Since it is locally small, we deduce that it is locally essentially small.
  • is locally finitely presentable
    Since it is finitary algebraic, we deduce that it is locally finitely presentable.
  • is additive
    Since it is abelian, we deduce that it is additive.
  • has equalizers
    Since it is abelian, we deduce that it has equalizers.
  • has coequalizers
    Since it is abelian, we deduce that it has coequalizers.
  • is mono-regular
    Since it is abelian, we deduce that it is mono-regular.
  • is epi-regular
    Since it is abelian, we deduce that it is epi-regular.
  • is Malcev
    Since it is abelian, we deduce that it is Malcev.
  • is balanced
    Since it is mono-regular, we deduce that it is balanced.
  • is finitely complete
    Since it is Malcev, we deduce that it is finitely complete.
  • is Cauchy complete
    Since it has coequalizers, we deduce that it is Cauchy complete.
  • is preadditive
    Since it is additive, we deduce that it is preadditive.
  • has finite coproducts
    Since it is additive, we deduce that it has finite coproducts.
  • has finite products
    Since it is finitely complete, we deduce that it has finite products.
  • has a terminal object
    Since it has finite products, we deduce that it has a terminal object.
  • has binary products
    Since it has finite products, we deduce that it has binary products.
  • has pullbacks
    Since it has binary products and has equalizers, we deduce that it has pullbacks.
  • is connected
    Since it has a terminal object, we deduce that it is connected.
  • is locally presentable
    Since it is locally finitely presentable, we deduce that it is locally presentable.
  • has exact filtered colimits
    Since it is locally finitely presentable, we deduce that it has exact filtered colimits.
  • is locally ℵ₁-presentable
    Since it is locally finitely presentable, we deduce that it is locally ℵ₁-presentable.
  • has zero morphisms
    Since it is preadditive, we deduce that it has zero morphisms.
  • has disjoint finite coproducts
    Since it is additive, we deduce that it has disjoint finite coproducts.
  • is inhabited
    Since it is connected, we deduce that it is inhabited.
  • is finitely cocomplete
    Since it has finite coproducts and has coequalizers, we deduce that it is finitely cocomplete.
  • has an initial object
    Since it has finite coproducts, we deduce that it has an initial object.
  • has binary coproducts
    Since it has finite coproducts, we deduce that it has binary coproducts.
  • has pushouts
    Since it has binary coproducts and has coequalizers, we deduce that it has pushouts.
  • is pointed
    Since it has zero morphisms and has a terminal object, we deduce that it is pointed.
  • has filtered colimits
    Since it has exact filtered colimits, we deduce that it has filtered colimits.
  • is well-powered
    Since it is locally presentable, we deduce that it is well-powered.
  • is well-copowered
    Since it is locally presentable, we deduce that it is well-copowered.
  • is complete
    Since it is locally presentable, we deduce that it is complete.
  • is cocomplete
    Since it is locally presentable, we deduce that it is cocomplete.
  • has a generator
    Since it is locally presentable, we deduce that it has a generator.
  • has wide pushouts
    Since it is cocomplete, we deduce that it has wide pushouts.
  • has connected colimits
    Since it is cocomplete, we deduce that it has connected colimits.
  • has coproducts
    Since it is cocomplete, we deduce that it has coproducts.
  • has countable coproducts
    Since it has coproducts, we deduce that it has countable coproducts.
  • has sequential colimits
    Since it has coequalizers and has countable coproducts, we deduce that it has sequential colimits.
  • has filtered limits
    Since it is complete, we deduce that it has filtered limits.
  • has wide pullbacks
    Since it is complete, we deduce that it has wide pullbacks.
  • has connected limits
    Since it is complete, we deduce that it has connected limits.
  • has products
    Since it is complete, we deduce that it has products.
  • has countable products
    Since it has products, we deduce that it has countable products.
  • has disjoint coproducts
    Since it has coproducts and has disjoint finite coproducts, we deduce that it has disjoint coproducts.
  • has sequential limits
    Since it has equalizers and has countable products, we deduce that it has sequential limits.
  • is Grothendieck abelian
    Since it is abelian and has coproducts and has a generator and has exact filtered colimits, we deduce that it is Grothendieck abelian.
  • has a cogenerator
    Since it is Grothendieck abelian, we deduce that it has a cogenerator.

Non-Properties

Non-Properties from the database

Deduced Non-Properties*

  • is not small
    Assume for a contradiction that it is small. Since it is small, we deduce that it is essentially small. Since it is essentially small and has products, we deduce that it is thin. Since it has disjoint finite coproducts and is thin, we deduce that it is trivial. Since it is trivial, we deduce that it is split abelian. This is a contradiction since we already know that it is not split abelian.
  • is not cartesian closed
    Assume for a contradiction that it is cartesian closed. Since it is pointed and is cartesian closed, we deduce that it is trivial. Since it is trivial, we deduce that it is split abelian. This is a contradiction since we already know that it is not split abelian.
  • is not an elementary topos
    Assume for a contradiction that it is an elementary topos. Since it is an elementary topos, we deduce that it is cartesian closed. This is a contradiction since we already know that it is not cartesian closed.
  • is not a Grothendieck topos
    Assume for a contradiction that it is a Grothendieck topos. Since it is a Grothendieck topos, we deduce that it is an elementary topos. This is a contradiction since we already know that it is not an elementary topos.
  • does not have a strict initial object
    Assume for a contradiction that it has a strict initial object. Since it has a strict initial object and is pointed, we deduce that it is trivial. Since it is trivial, we deduce that it is a Grothendieck topos. This is a contradiction since we already know that it is not a Grothendieck topos.
  • does not have a strict terminal object
    Assume for a contradiction that it has a strict terminal object. Since it has a strict terminal object and is pointed, we deduce that it is trivial. Since it is trivial, we deduce that it is a Grothendieck topos. This is a contradiction since we already know that it is not a Grothendieck topos.
  • is not self-dual
    Assume for a contradiction that it is self-dual. Since it is locally presentable and is self-dual, we deduce that it is thin. Since it is thin, we deduce that it is right cancellative. Since it is right cancellative and has a terminal object, we deduce that it has a strict terminal object. This is a contradiction since we already know that it does not have a strict terminal object.
  • is not a groupoid
    Assume for a contradiction that it is a groupoid. Since it is a groupoid, we deduce that it is self-dual. This is a contradiction since we already know that it is not self-dual.
  • is not essentially small
    Assume for a contradiction that it is essentially small. Since it is essentially small and has products, we deduce that it is thin. Since it is thin, we deduce that it is left cancellative. Since it is left cancellative and has an initial object, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
  • is not thin
    Assume for a contradiction that it is thin. Since it is thin, we deduce that it is left cancellative. Since it is left cancellative and has an initial object, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
  • is not discrete
    Assume for a contradiction that it is discrete. Since it is discrete, we deduce that it is skeletal. This is a contradiction since we already know that it is not skeletal.
  • is not essentially discrete
    Assume for a contradiction that it is essentially discrete. Since it is essentially discrete, we deduce that it is thin. This is a contradiction since we already know that it is not thin.
  • is not finite
    Assume for a contradiction that it is finite. Since it is finite, we deduce that it is small. This is a contradiction since we already know that it is not small.
  • is not essentially finite
    Assume for a contradiction that it is essentially finite. Since it is essentially finite and has finite products, we deduce that it is thin. This is a contradiction since we already know that it is not thin.
  • is not trivial
    Assume for a contradiction that it is trivial. Since it is trivial, we deduce that it is a Grothendieck topos. This is a contradiction since we already know that it is not a Grothendieck topos.
  • does not have a subobject classifier
    Assume for a contradiction that it has a subobject classifier. Since it is additive and has pullbacks and has a subobject classifier, we deduce that it is trivial. This is a contradiction since we already know that it is not trivial.
  • is not infinitary distributive
    Assume for a contradiction that it is infinitary distributive. Since it is infinitary distributive, we deduce that it is distributive. Since it is distributive, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
  • is not distributive
    Assume for a contradiction that it is distributive. Since it is distributive, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.
  • is not left cancellative
    Assume for a contradiction that it is left cancellative. Since it is left cancellative and has coequalizers, we deduce that it is thin. This is a contradiction since we already know that it is not thin.
  • is not right cancellative
    Assume for a contradiction that it is right cancellative. Since it is right cancellative and has an initial object, we deduce that it has a strict initial object. This is a contradiction since we already know that it does not have a strict initial object.

*This also uses the deduced properties.

Unknown properties

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Special morphisms

  • Isomorphisms: bijective RR-linear maps
    This characterization holds in every algebraic category.
  • Monomorphisms: injective RR-linear maps
  • Epimorphisms: surjective RR-linear maps
    The forgetful functor to abelian groups is faithful and preserves colimits, hence reflects and preserves epimorphisms. Alternatively, just use the same proof as for abelian groups.