Semiprojectivity and semiinjectivity in different categories

Abstract

Projectivity and injectivity are fundamental notions in category theory. We consider natural weakenings termed semiprojectivity and semiinjectivity, and study these concepts in different categories. For example, in the category of metric spaces, (semi)injective objects are precisely the absolute (neighborhood) retracts. We show that the trivial group is the only semiinjective group, while every free product of a finitely presented group and a free group is semiprojective. To a compact, metric space $X$ we associate the commutative C*-algebra $C(X)$. This association is contravariant, whence semiinjectivity of $X$ is related to semiprojectivity of $C(X)$. Together with Adam S{\o}rensen, we showed that $C(X)$ is semiprojective in the category of all C*-algebras if and only if $X$ is an absolute neighborhood retract with dimension at most one.