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
Assigned properties
- is locally small
- is semi-strongly connected
- has a generator
- has a cogenerator
- is well-powered
- is well-copowered
- is complete
- is cocomplete
- is infinitary extensive
- has filtered-colimit-stable monomorphisms
- has a regular subobject classifier
Deduced properties
- has connected limits
- is finitely complete
- has equalizers
- has products
- is multi-complete
- is connected
- has filtered colimits
- has coproducts
- is extensive
- has a generating set
- is inhabited
- is locally essentially small
- has connected colimits
- is finitely cocomplete
- has coequalizers
- is multi-cocomplete
- has a cogenerating set
- has finite products
- has a multi-terminal object
- has coreflexive equalizers
- is Cauchy complete
- has finite coproducts
- has disjoint finite coproducts
- has a strict initial object
- is filtered
- has directed colimits
- has ℵ₁-filtered colimits
- has sifted colimits
- has powers
- has ℵ₂-small products
- has wide pullbacks
- has a multi-initial object
- has reflexive coequalizers
- is cofiltered
- has cosifted limits
- has copowers
- has ℵ₂-small coproducts
- has wide pushouts
- has a terminal object
- has quotients of congruences
- has disjoint coproducts
- is distributive
- is infinitary distributive
- is sifted
- has an initial object
- has countable products
- has ℵ₂-small powers
- has binary products
- has finite powers
- has cofiltered limits
- has pullbacks
- has coquotients of cocongruences
- is cosifted
- has sequential colimits
- has countable coproducts
- has ℵ₂-small copowers
- has binary coproducts
- has finite copowers
- has pushouts
- is countably distributive
- has cocartesian cofiltered limits
- is ℵ₁-filtered
- has sequential limits
- has countable powers
- has binary powers
- is ℵ₁-cofiltered
- has directed limits
- has ℵ₁-cofiltered limits
- has countable copowers
- has binary copowers
- has a natural numbers object
Unsatisfied Properties
Assigned properties
- is not skeletal
- is not balanced
- is not Malcev
- does not have cartesian filtered colimits
- does not have cofiltered-limit-stable epimorphisms
- does not have effective cocongruences
- is not regular
Deduced properties*
- is not thin
- is not additive
- is not abelian
- is not locally strongly finitely presentable
- is not left cancellative
- is not cartesian closed
- is not locally cartesian closed
- is not Barr-exact
- is not discrete
- does not have exact filtered colimits
- does not have biproducts
- is not mono-regular
- is not gaunt
- is not direct
- is not right cancellative
- is not core-thin
- is not Barr-coexact
- does not have exact cofiltered limits
- is not coextensive
- is not quotient-trivial
- is not essentially finite
- is not epi-regular
- is not inverse
- is not self-dual
- is not locally finitely presentable
- is not a generalized variety
- is not preadditive
- is not Grothendieck abelian
- is not split abelian
- is not finitary algebraic
- does not have effective congruences
- is not trivial
- is not essentially discrete
- does not have a strict terminal object
- is not a groupoid
- is not normal
- is not finite
- does not have a subobject classifier
- is not subobject-trivial
- is not locally finite
- is not one-way
- is not essentially small
- is not essentially countable
- is not an elementary topos
- is not a Grothendieck topos
- is not a pretopos
- is not cocartesian coclosed
- does not have disjoint finite products
- is not infinitary coextensive
- is not conormal
- does not have a quotient object classifier
- does not have a regular quotient object classifier
- is not pointed
- is not finitely accessible
- is not multi-algebraic
- is not strongly connected
- is not small
- is not countable
- is not locally cocartesian coclosed
- does not have disjoint products
- is not codistributive
- is not unital
- is not locally finitely multi-presentable
- does not have zero morphisms
- is not counital
- is not countably codistributive
- does not have kernels
- does not satisfy CIP
- is not infinitary codistributive
- does not have cokernels
- does not satisfy CSP
*This also uses the deduced satisfied properties.
Unknown properties
There are 10 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: A measurable map is a regular epimorphism iff is surjective and is a quotient map, meaning that a subset of is measurable when its -preimage is measurable.
Comments
- The thread MSE/5024471 asks for the finitely presentable objects of this category.