category of measurable spaces
- notation:
- objects: measurable spaces
- morphisms: measurable maps
- Related categories:
- nLab Link
This is very similar to the category of topological spaces. Accordingly, limits and colimits can be constructed in the same way.
Satisfied Properties
Properties from the database
- is cocomplete
- has a cogenerator
- is complete
- has a generator
- is infinitary extensive
- is locally small
- has a regular subobject classifier
- is semi-strongly connected
- 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 generating set
- is inhabited
- has a natural numbers object
- 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
- has a cogenerating set
- is cosifted
- is cofiltered
Unsatisfied Properties
Properties from the database
- is not balanced
- does not have cartesian filtered colimits
- is not Malcev
- is not skeletal
Deduced properties*
- does not have biproducts
- does not have exact filtered colimits
- is not mono-regular
- is not a groupoid
- is not normal
- is not direct
- is not additive
- is not preadditive
- 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
- is not right cancellative
- is not left cancellative
- is not one-way
- is not essentially small
- is not small
- is not essentially countable
- is not essentially finite
- is not finite
- is not countable
- is not locally finitely presentable
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not finitely accessible
- is not a generalized variety
- is not locally finitely multi-presentable
- is not multi-algebraic
- is not cartesian closed
- is not an elementary topos
- does not have a subobject classifier
- is not locally cartesian closed
- 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 coextensive
- is not infinitary coextensive
- is not codistributive
- is not countably codistributive
- is not infinitary codistributive
- is not cocartesian coclosed
- 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 11 properties for which the database doesn't have an answer if they are satisfied or not. Please help to contribute the data!
Special objects
- terminal object: singleton set with the unique -algebra
- initial object: empty set with the unique -algebra
- products: direct products with the product -algebra
- coproducts: disjoint union with the obvious -algebra
Special morphisms
- isomorphisms: bijective measurable maps that map measurable sets to measurable sets
- monomorphisms: injective measurable maps
- epimorphisms: surjective measurable maps
- regular monomorphisms: embeddings
- regular epimorphisms:
Comments
- The thread MSE/5024471 asks for the finitely presentable objects of this category.