category of locally ringed spaces
- notation:
- objects: locally ringed spaces
- morphisms: morphisms of locally ringed spaces, thus consisting of a continuous map and a homomorphism of sheaves that induces local ring homomorphisms in the stalks
- Related categories:
- nLab Link
Satisfied Properties
Properties from the database
- is cocomplete
- is complete
- is infinitary extensive
- is locally small
- is well-copowered
- is well-powered
Deduced properties
- has connected limits
- is finitely complete
- has equalizers
- has coreflexive equalizers
- has products
- has countable products
- has finite products
- has powers
- has binary products
- has a terminal object
- has countable powers
- has pullbacks
- is Cauchy complete
- has wide pullbacks
- has cofiltered limits
- has sequential limits
- has finite powers
- has binary powers
- is multi-complete
- has a multi-terminal object
- is connected
- has coproducts
- is extensive
- has finite coproducts
- has cocartesian cofiltered limits
- has disjoint finite coproducts
- has disjoint coproducts
- has a strict initial object
- has an initial object
- is distributive
- is infinitary distributive
- is countably distributive
- has countable coproducts
- is locally essentially small
- has a natural numbers object
- is inhabited
- is filtered
- is sifted
- has connected colimits
- has sifted colimits
- has filtered colimits
- has reflexive coequalizers
- has directed colimits
- is finitely cocomplete
- has cosifted limits
- has copowers
- has binary coproducts
- has coequalizers
- has countable copowers
- has sequential colimits
- has pushouts
- has directed limits
- has wide pushouts
- has finite copowers
- has binary copowers
- is multi-cocomplete
- has a multi-initial object
- is cosifted
- is cofiltered
Unsatisfied Properties
Properties from the database
- is not balanced
- is not co-Malcev
- is not Malcev
- is not semi-strongly connected
- is not skeletal
Deduced properties*
- is not mono-regular
- is not a groupoid
- is not normal
- is not direct
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not discrete
- is not essentially discrete
- is not trivial
- is not thin
- does not have a strict terminal object
- is not pointed
- does not have zero morphisms
- does not have kernels
- does not have biproducts
- is not right cancellative
- is not left cancellative
- is not one-way
- is not preadditive
- is not additive
- is not essentially small
- is not small
- is not essentially countable
- is not essentially finite
- is not finite
- is not countable
- does not have a subobject classifier
- is not an elementary topos
- is not a Grothendieck topos
- is not strongly connected
- is not unital
- does not have cokernels
- does not have disjoint finite products
- does not have disjoint products
- is not codistributive
- is not countably codistributive
- is not infinitary codistributive
- is not cocartesian coclosed
- is not coextensive
- is not infinitary coextensive
- is not epi-regular
- is not conormal
- is not inverse
- does not have a quotient object classifier
- does not have a regular quotient object classifier
- is not locally cocartesian coclosed
- is not counital
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
There are 26 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!
- is accessible
- is cartesian closed
- has cartesian filtered colimits
- is coaccessible
- has a cogenerating set
- has a cogenerator
- is coregular
- has exact filtered colimits
- is finitary algebraic
- is finitely accessible
- is a generalized variety
- has a generating set
- has a generator
- is locally cartesian closed
- is locally copresentable
- is locally finitely multi-presentable
- is locally finitely presentable
- is locally multi-presentable
- is locally poly-presentable
- is locally presentable
- is locally strongly finitely presentable
- is locally ℵ₁-presentable
- is multi-algebraic
- is regular
- has a regular subobject classifier
- is ℵ₁-accessible
Special objects
- terminal object:
- initial object: empty space
- products: See Localization of ringed spaces by W. Gillam. See also MSE/1033675.
- coproducts: disjoint union with the product sheaf
Special morphisms
- isomorphisms: pairs consisting of a homeomorphism and an isomorphism of sheaves
- monomorphisms:
- epimorphisms:
- regular monomorphisms:
- regular epimorphisms: