sequences functor on sets

notation: $(-)^{\IN}$ : $\Set$ → $\Set$
Source: category of sets
Target: category of sets
Left adjoint: $\IN \times (-)$
Related functors: $\times$ , $\id_{\Set}$ , $(-)^2$

This functor maps a set $X$ to the countable power $X^{\IN}$ , i.e. the set of sequences in $X$ . It is an example of a polynomial functor. It is also an example of a monadic functor for which the crude monadicity theorem does not apply.

Satisfied Properties

Assigned properties

preserves epimorphisms
preserves initial objects
is monadic
is representable

Deduced properties

is continuous
is conservative
is faithful
is a right adjoint
is cofinitary
is left exact
preserves products
preserves finite products
preserves equalizers
preserves monomorphisms
preserves terminal objects
preserves coreflexive equalizers

Unsatisfied Properties

Assigned properties

is not essentially surjective
is not essentially injective
is not finitary
does not preserve finite coproducts
does not preserve reflexive coequalizers

Deduced properties*

is not an equivalence
is not full
is not left-invertible
is not cocontinuous
does not preserve coproducts
is not right exact
does not preserve coequalizers
is not exact
is not a left adjoint
is not comonadic

*This also uses the deduced satisfied properties.

Unknown properties

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