category of measurable spaces
- Notation:
- Objects: measurable spaces
- Morphisms: measurable maps
- Related categories: ,
- nLab Link
This category is similar to the category of topological spaces. For instance, limits and colimits can be constructed in the same way. However, a main difference is that this category is not infinitary distributive.
Satisfied Properties
Assigned properties
- is locally small
- is semi-strongly connected
- has a generator
- is well-powered
- is well-copowered
- is complete
- is cocomplete
- is coregular
- is countably extensive
- has filtered-colimit-stable monomorphisms
- has a regular subobject classifier
- has an extremal cogenerator
- is co-Malcev
Deduced properties
- has connected limits
- is finitely complete
- has equalizers
- has products
- is multi-complete
- is connected
- has filtered colimits
- has countable coproducts
- is extensive
- has a generating set
- is inhabited
- is locally essentially small
- is finitely cocomplete
- has connected colimits
- has coequalizers
- has coproducts
- is multi-cocomplete
- has a cogenerator
- has an extremal 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
- is ℵ₁-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 sequential colimits
- has cosifted limits
- has a cogenerating set
- has copowers
- has ℵ₂-small coproducts
- has countable copowers
- has wide pushouts
- has a terminal object
- has quotients of congruences
- has disjoint coproducts
- is distributive
- is countably 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 ℵ₂-small copowers
- has binary coproducts
- has finite copowers
- has pushouts
- has a natural numbers object
- has cocartesian cofiltered limits
- has sequential limits
- has countable powers
- has binary powers
- is ℵ₁-cofiltered
- has directed limits
- has ℵ₁-cofiltered limits
- has binary copowers
Unsatisfied Properties
Assigned properties
- is not skeletal
- does not have cofiltered-limit-stable epimorphisms
- does not have effective cocongruences
- is not regular
- does not have an extremal generating set
- is not infinitary distributive
- is not coaccessible
Deduced properties*
- is not accessible
- is not abelian
- is not finitary algebraic
- is not left cancellative
- is not locally cartesian closed
- is not Barr-exact
- is not discrete
- is not cartesian closed
- does not have cartesian filtered colimits
- is not infinitary extensive
- does not have an extremal generator
- is not balanced
- is not a groupoid
- is not essentially small
- is not gaunt
- is not direct
- is not thin
- is not locally copresentable
- is not right cancellative
- is not core-thin
- is not Barr-coexact
- does not have exact cofiltered limits
- is not quotient-trivial
- is not essentially finite
- is not inverse
- is not self-dual
- is not locally presentable
- is not ℵ₁-accessible
- is not a generalized variety
- is not locally multi-presentable
- is not locally poly-presentable
- is not Grothendieck abelian
- is not split abelian
- is not one-sorted finitary algebraic
- is not core-connected
- is not trivial
- is not essentially discrete
- does not have a strict terminal object
- does not have exact filtered colimits
- does not have biproducts
- is not mono-regular
- is not small
- is not finite
- is not essentially countable
- is not regular-subobject-trivial
- is not locally finite
- is not one-way
- is not an elementary topos
- is not a Grothendieck topos
- is not a pretopos
- is not a quasitopos
- is not cocartesian coclosed
- does not have disjoint finite products
- is not coextensive
- is not epi-regular
- does not have a regular quotient object classifier
- is not regular-quotient-trivial
- is not pointed
- is not locally ℵ₁-presentable
- is not locally finitely presentable
- is not finitely accessible
- is not additive
- is not multi-algebraic
- does not have effective congruences
- is not strongly connected
- is not normal
- is not countable
- does not have a subobject classifier
- is not subobject-trivial
- is not Malcev
- is not locally cocartesian coclosed
- does not have disjoint products
- is not codistributive
- is not countably coextensive
- is not conormal
- does not have a quotient object classifier
- is not unital
- is not locally finitely multi-presentable
- is not preadditive
- does not have zero morphisms
- is not counital
- is not countably codistributive
- is not infinitary coextensive
- 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
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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.