category of schemes
- notation:
- objects: schemes
- morphisms: morphisms of locally ringed spaces
- Related categories: ,
- nLab Link
Satisfied Properties
Properties from the database
- is infinitary extensive
- is locally small
- has pullbacks
- has a terminal object
- is well-powered
Deduced properties
- has binary products
- has finite products
- is connected
- has equalizers
- has coreflexive equalizers
- is finitely complete
- is Cauchy complete
- has finite powers
- has binary powers
- has coproducts
- is extensive
- has finite coproducts
- has a strict initial object
- has an initial object
- has disjoint finite coproducts
- has disjoint coproducts
- is distributive
- is infinitary distributive
- is lextensive
- is locally essentially small
- is inhabited
- has countable coproducts
- has binary coproducts
- has copowers
- has countable copowers
- has finite copowers
- has binary copowers
Unsatisfied Properties
Properties from the database
- is not balanced
- does not have countable powers
- does not have a generating set
- is not Malcev
- does not have pushouts
- is not skeletal
- does not have a strict terminal object
- is not strongly connected
Deduced properties*
- does not have countable products
- does not have products
- does not have cofiltered limits
- is not complete
- does not have wide pullbacks
- does not have connected limits
- is not essentially finite
- does not have sequential limits
- does not have directed limits
- does not have powers
- is not cartesian closed
- is not finite
- does not have a generator
- is not essentially small
- is not small
- is not finitary algebraic
- is not a groupoid
- is not mono-regular
- is not direct
- is not discrete
- does not have zero morphisms
- does not have biproducts
- is not pointed
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not essentially discrete
- is not trivial
- is not thin
- is not left cancellative
- is not one-way
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not locally strongly finitely presentable
- is not an elementary topos
- does not have a subobject classifier
- is not locally cartesian closed
- is not a Grothendieck topos
- is not unital
- does not have cosifted limits
- does not have coequalizers
- does not have reflexive coequalizers
- does not have sifted colimits
- does not have connected colimits
- is not cocomplete
- is not finitely cocomplete
- does not have wide pushouts
- does not have disjoint products
- does not have disjoint finite products
- is not infinitary codistributive
- is not right cancellative
- is not infinitary coextensive
- is not codistributive
- is not coextensive
- is not coregular
- is not epi-regular
- is not inverse
- is not co-Malcev
- is not counital
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
There are 9 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!
- has a cogenerating set
- has a cogenerator
- has directed colimits
- has exact filtered colimits
- has filtered colimits
- is regular
- has a regular subobject classifier
- has sequential colimits
- is well-copowered
Special objects
- terminal object:
- initial object: empty scheme
- products: [finite case] The idea is to use and then to glue affine pieces together. See EGA I, Chap. I, Thm. 3.2.1.
- 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: