free group functor

Notation

F_{\Grp}: \Set \to \Grp

Domain category of setsCodomain category of groupsRight adjoint forgetful functor for groups Related

F_{\Mon,\Grp}

,

\IZ[-]

External nLab Link

This functor maps a set $X$ to the free group $F_{\Grp}(X)$ on that set. In the proofs, we abbreviate $F \coloneqq F_{\Grp}$ .

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

is not full on isomorphisms
is not dominant
does not preserve terminal objects
does not preserve equalizers
is not cofinitary
is not left-invertible

Deduced properties*

is not a reflector
is not an equivalence
is not continuous
does not preserve finite products
is not left exact
does not preserve binary products
is not full
is not pseudomonic
is not essentially surjective
is not fully faithful
is not a right adjoint
is not an isomorphism
does not preserve products
is not exact
is not representable
is not regular
is not right-invertible
is not monadic
is not a coreflector

*This also uses the deduced satisfied properties.

Unknown properties

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