trivial category
This is the simplest category, consisting of a single object and its identity morphism . A concrete representation is the full subcategory of consisting of the empty set. It is the terminal object in the category of small categories.
Satisfied properties
Assigned properties
Deduced properties
- is core-connected
- is a Grothendieck topos
- is essentially discrete
- is essentially finite
- is one-sorted finitary algebraic
- is self-dual
- is split abelian
- is direct
- is locally small
- is essentially small
- is countable
- is a Grothendieck cotopos
- is dually one-sorted finitary algebraic
- is inverse
- is abelian
- is finitary algebraic
- is well-copowered
- is a dual generalized variety
- is finitely coaccessible
- is strongly connected
- has an extremal generator
- is a groupoid
- is thin
- has connected limits
- is locally essentially small
- has cosifted limits
- has an extremal generating collection
- is well-powered
- is essentially countable
- is locally finite
- is one-way
- is skeletal
- is cartesian closed
- has exact filtered colimits
- is locally presentable
- has a subobject classifier
- is dually finitary algebraic
- is finitely accessible
- is a generalized variety
- has an extremal cogenerator
- has connected colimits
- has sifted colimits
- has an extremal cogenerating collection
- is cocartesian coclosed
- has exact cofiltered limits
- is locally copresentable
- has a quotient object classifier
- is accessible
- is cocomplete
- has filtered-colimit-stable monomorphisms
- is locally finitely presentable
- is ℵ₁-accessible
- has filtered colimits
- is complete
- is locally finitely multi-presentable
- is additive
- has cokernels
- is conormal
- has kernels
- is normal
- has pullback-stable regular epimorphisms
- has effective congruences
- is multi-algebraic
- has finite products
- is locally cartesian closed
- is semi-strongly connected
- is finitely complete
- has cartesian filtered colimits
- has reflexive coequalizers
- has a generator
- has a generating collection
- has directed limits
- is left cancellative
- has equalizers
- has wide pullbacks
- is mono-regular
- has a regular subobject classifier
- is subobject-trivial
- is retract-trivial
- is Cauchy complete
- is core-thin
- has binary powers
- is coaccessible
- has cofiltered-limit-stable epimorphisms
- is locally finitely copresentable
- is ℵ₁-coaccessible
- has cofiltered limits
- is locally finitely multi-copresentable
- has pushout-stable regular monomorphisms
- has effective cocongruences
- is dually multi-algebraic
- has finite coproducts
- is locally cocartesian coclosed
- is finitely cocomplete
- has cocartesian cofiltered limits
- has coreflexive equalizers
- has a cogenerator
- has a cogenerating collection
- has directed colimits
- is right cancellative
- has coequalizers
- has wide pushouts
- is epi-regular
- has a regular quotient object classifier
- is quotient-trivial
- has binary copowers
- is Malcev
- has a parametrized natural numbers object
- is locally ℵ₁-presentable
- has ℵ₁-filtered colimits
- is locally multi-presentable
- is multi-cocomplete
- is locally poly-presentable
- is preadditive
- has biproducts
- has kernel pairs
- has pullbacks
- has products
- is multi-complete
- has quotients of congruences
- is connected
- has disjoint finite products
- is distributive
- has zero morphisms
- has a multi-initial object
- has a multi-terminal object
- is filtered
- has sequential limits
- is inhabited
- is balanced
- has binary products
- has a terminal object
- has finite powers
- is concretizable
- is regular-subobject-trivial
- is gaunt
- is an elementary topos
- is a quasitopos
- is total
- is co-Malcev
- has a coparametrized conatural numbers object
- is locally ℵ₁-copresentable
- has ℵ₁-cofiltered limits
- is locally multi-copresentable
- is locally poly-copresentable
- has cokernel pairs
- has pushouts
- has coproducts
- has coquotients of cocongruences
- has disjoint finite coproducts
- is codistributive
- is cofiltered
- has sequential colimits
- has binary coproducts
- has an initial object
- has finite copowers
- is regular-quotient-trivial
- is an elementary cotopos
- is a quasicotopos
- is cototal
- is unital
- has a natural numbers object
- is Grothendieck abelian
- has a strict initial object
- has coequalizers of kernel pairs
- is pointed
- has disjoint coproducts
- is infinitary distributive
- satisfies CIP
- is extensive
- is sifted
- is ℵ₁-filtered
- has powers
- has ℵ₂-small products
- has countable products
- has countable powers
- has copowers
- is coregular
- is counital
- has a conatural numbers object
- is co-Grothendieck abelian
- has a strict terminal object
- has equalizers of cokernel pairs
- has disjoint products
- is infinitary codistributive
- satisfies CSP
- is coextensive
- is cosifted
- is ℵ₁-cofiltered
- has ℵ₂-small coproducts
- has countable coproducts
- has countable copowers
- is regular
- is countably distributive
- is infinitary extensive
- is countably extensive
- has ℵ₂-small powers
- is Barr-exact
- is countably codistributive
- is infinitary coextensive
- is countably coextensive
- has ℵ₂-small copowers
- is Barr-coexact
- is a pretopos
- is a precotopos
Unsatisfied properties
Assigned properties
—
Deduced properties*
—
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- terminal object: the unique object
- initial object: the unique object
- products:
- coproducts:
Special morphisms
- isomorphisms: every morphism
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: every morphism
- regular epimorphisms: every morphism
Functors
The database stores 5 functors based on the trivial category.
- trivial functor from the category of groups
- trivial functor from the category of sets
- trivial functor from the delooping
- trivial functor from the walking idempotent
- walking isomorphism object inclusion
Symmetric monoidal categories
The database stores 1 symmetric monoidal category based on the trivial category.