walking span
- notation:
- objects:
- morphisms: , , and the identities
- Related categories: ,
- nLab Link
The name of this category comes from the fact that a functor out of it is the same as a span in the target category. It is isomorphic to the partial order of proper positive divisors of .
Satisfied Properties
Properties from the database
- has binary products
- has directed colimits
- is finite
- has an initial object
- is locally cartesian closed
- is skeletal
- is thin
Deduced properties
- has filtered colimits
- is essentially finite
- is small
- is essentially small
- is locally small
- is locally essentially small
- is well-copowered
- is well-powered
- has a generating set
- has equalizers
- has pullbacks
- has wide pullbacks
- has connected limits
- has cofiltered limits
- is Cauchy complete
- is left cancellative
- has a strict initial object
- is connected
- is inhabited
- has a generator
- has sequential colimits
- has directed limits
- has sequential limits
- has a cogenerating set
- has coequalizers
- is right cancellative
- has a cogenerator
Unsatisfied Properties
Properties from the database
- does not have binary coproducts
- is not strongly connected
- does not have a terminal object
Deduced properties*
- does not have finite products
- does not have products
- does not have countable products
- is not complete
- is not finitely complete
- does not have biproducts
- does not have exact filtered colimits
- is not infinitary distributive
- is not distributive
- is not regular
- is not lextensive
- is not a groupoid
- is not balanced
- is not mono-regular
- does not have zero morphisms
- is not pointed
- is not preadditive
- is not additive
- is not abelian
- is not Grothendieck abelian
- is not split abelian
- is not locally strongly finitely presentable
- is not finitary algebraic
- is not essentially discrete
- is not discrete
- is not trivial
- does not have disjoint finite coproducts
- does not have disjoint coproducts
- is not extensive
- is not infinitary extensive
- is not locally presentable
- is not locally finitely presentable
- is not locally ℵ₁-presentable
- is not cartesian closed
- is not an elementary topos
- does not have a subobject classifier
- does not have a regular subobject classifier
- is not a Grothendieck topos
- is not Malcev
- is not unital
- does not have finite coproducts
- does not have coproducts
- does not have pushouts
- does not have countable coproducts
- is not cocomplete
- is not finitely cocomplete
- does not have wide pushouts
- does not have connected colimits
- does not have disjoint products
- does not have disjoint finite products
- is not infinitary codistributive
- is not codistributive
- does not have a strict terminal object
- is not coextensive
- is not infinitary coextensive
- is not coregular
- is not epi-regular
- is not co-Malcev
- is not counital
- is not self-dual
*This also uses the deduced satisfied properties.
Unknown properties
—
Special objects
- initial object:
- products: [binary case] , ,
Special morphisms
- isomorphisms: the three identities
- monomorphisms: every morphism
- epimorphisms: every morphism
- regular monomorphisms: same as isomorphisms
- regular epimorphisms: same as isomorphisms