Controlled objects in left-exact $\infty$-categories and the Novikov conjecture

Abstract

We associate to every G-bornological coarse space X and every left-exact ∞-category with G-action a left-exact infinity-category of equivariant X-controlled objects. Postcomposing with algebraic K-theory leads to new equivariant coarse homology theories. This allows us to apply the injectivity results for assembly maps by Bunke, Engel, Kasprowski and Winges to the algebraic K-theory of left-exact ∞-categories. ContentsC 3.1. Coarse invariance 3.2. Flasques 3.3. u-continuity 3.4. Subspace inclusions 3.5. Excision 3.6. Strong additivity 4. Further constructions 4.1. Forcing continuity 4.2. Colimits 4.3. V G,c and V c G 4.4. Transfers 5. Calculations 6. Equivariant coarse homology theories 6.1. Basic definitions 6.2. Homological functors 6.3. CP-functors Date: November 11, 2019. 1 2 U. BUNKE, D.-C. CISINSKI, D. KASPROWSKI, AND C. WINGES 6.4. Algebraic K-theory 97 6.5. Split injectivity results 98 7. ∞-category background 101 7.1. Left-exact ∞-categories 101 7.2. The calculus of fractions formula 112 7.3. Labellings and localisation 114 7.4. Stabilisation and cofibres 116 7.5. Excisive squares in Cat Lex ∞, * 119 7.6. The universal property of the bounded derived category 121 7.7. A-theory as a GOrb-spectrum 131 References 134

On the Farrell-Jones conjecture for localising invariants

“…If H is a finitary localising invariant (see Definition 1.2), then we know by [BCKW,Cor. 6.18] that HV c,perf…”

“…Let X be a G-bornological coarse space and (Y, Z) be a partition of X into invariant and coarsely disjoint subsets. Since Cat Lex,perf ∞, * is semi-additive [BCKW,Lem. 7.21] and V c C is π 0 -excisive [BCKW,Lem.…”

“…This definition generalises [BGT13] from stable to left-exact ∞-categories. In fact, any finitary localising invariant on left-exact ∞-categories arises from a localising invariant on stable ∞-categories in the sense of [BGT13] by precomposing with the stabilisation functor [BCKW,Lem. 6.9].…”

“…As explained in [BCKW,Sec. 6.2], finitary localising invariants arise by precomposing finitary, localising invariants on Cat ex ∞ with the stabilisation functor Stab : Cat Lex ∞, * → Cat ex ∞ .…”

“…For a ring R, we could take the additive category A = Mod(R) fg,proj of finitely generated projective R-modules with the trivial G-action. The generalisation from discrete rings to associative ring spectra R is obtained by replacing Mod(R) fg,proj with the left-exact ∞-category Mod(R) perf,op , the opposite of the ∞-category of perfect R-modules, see [BCKW,Ex. 1.4].…”