Homotopy Theory for C^{*}-algebras

Category of fibrant objects is a convenient framework to do homotopy theory, introduced and developed by Ken Brown. In this paper, we apply it to the category of C^{}-algebras. In particular, we get a unified treatment of (ordinary) homotopy theory for C^{}-algebras, KK-theory and E-theory, as all of these can be expressed as the homotopy category of a category of fibrant objects.

💡 Research Summary

The paper presents a unified homotopy‑theoretic framework for C*‑algebras by applying Ken Brown’s notion of a “category of fibrant objects” (CFO) to the non‑commutative setting. After a concise motivation—standard model‑category structures are difficult to impose on C*‑algebras—the author defines a CFO structure on the category of C*‑algebras as follows. Objects are all C*‑algebras, which are automatically fibrant because every *‑homomorphism satisfies the required lifting properties. Weak equivalences are taken to be *‑homotopy equivalences: two *‑homomorphisms f,g : A→B are weakly equivalent if there exists a continuous path H : A→C(