2018/02/14 by Hannes Thiel, Thiel, Hannes
Mathematics · #06F05 #18A20 #20E05 #46L05 #54C55 #55M15 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #Category Theory (math.CT) #FOS: Mathematics #General Topology (math.GN) #Group Theory (math.GR) #Operator Algebras (math.OA) #Primary 18A05 #Secondary 06B35
paper · pdf · doi:10.48550/arxiv.1802.05037
openalex publication_date 2018/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Sørensen, 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.