Homotopy colimits in stable representation theory

Abstract

We study the problem of existence and uniqueness of homotopy colimits in stable representation theory, where one typically does not have model category structures to guarantee that these homotopy colimits exist or have good properties. We get both negative results (homotopy cofibers fail to exist if there exist any objects of positive finite projective dimension!) and positive results (reasonable conditions under which homotopy colimits exist and are unique, even when model category structures fail to exist). We describe some applications to Waldhausen K-theory and to deformation-theoretic methods in stable representation theory.

Relative homological algebra and Waldhausen $K$-theory

“…C/E (A, −) for all i ≥ 1; this is Lemma 3.6 of [11]. By the Five Lemma, we then have a natural isomorphism of functors Ext 2…”

“…The horizontal maps marked as isomorphisms are isomorphisms because an Estable equivalence A → B induces a natural equivalence of functors Ext i C/E (B, −) −→ Ext i C/E (A, −) for all i ≥ 1; this is Lemma 3.6 of [11]. By the Five Lemma, we then have a natural isomorphism of functors Ext 2…”

“…• Axiom (Weq 2) is actually fairly substantial and takes some work to proveenough so that we moved this work into a paper of its own, [11]. In Corollary 4.4 of that paper, we prove that, if E = F, C has enough E-projectives and enough Einjectives, E has sectile epics and retractile monics, and every E-projective object is E-injective, then C satisfies axiom (Weq 2).…”

“…But since every E-projective is E-injective, this natural transformation being a natural isomorphism implies that Y → Y ′ is an E-stable equivalence; this is Lemma 4.2 of [11]. Hence the extension axiom is satisfied if E = F. Now if we do not have E = F but instead F ⊆ E, then the extension axiom remains satisfied, as we have fewer diagrams to check the extension axiom for.…”