functor of continuous functions

notation: $C$ : $\Top^{\op}$ → $\CAlg(R)$
Source: dual of the category of topological spaces
Target: category of commutative algebras

This functor maps a topological space $X$ to the commutative $\IR$ -algebra $C(X)$ of continuous functions $X \to \IR$ . A continuous map $f : X \to Y$ is mapped to the algebra homomorphism $f^* : C(Y) \to C(X)$ , $u \mapsto u \circ f$ .

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

is not essentially injective
is not essentially surjective
is not faithful
is not full
does not preserve finite coproducts
does not preserve reflexive coequalizers
does not preserve epimorphisms
is not finitary
is not representable

Deduced properties*

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

*This also uses the deduced satisfied properties.

Unknown properties

—