Semiprojectivity and semiinjectivity in different categories

TL;DR

This paper explores weakened forms of projectivity and injectivity in various categories, linking topological properties of spaces to algebraic properties of associated C*-algebras, and establishing new results on their structure and classification.

Contribution

It introduces and studies semiprojectivity and semiinjectivity across different categories, revealing new characterizations and relationships, especially between topological spaces and C*-algebras.

Findings

Only the trivial group is semiinjective.

Finitely presented groups combined with free groups are semiprojective.

C(X) is semiprojective iff X is an ANR with dimension ≤ 1.

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.