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This Dissertation was approved for publication on 2017-07-13 at 15:08. McCarthy, Randy Ando, Matthew Rezk, Charles Malkiewich, Cary Villeta-Garcia, Juan S 2017-09-29T17:52:21Z 2017-09-29T17:52:21Z 2019-09-30T09:15:23Z 2017-07-13 2017-08 We define and study the $K$-theory of exact categories with coefficients in endofunctors of spectra in analogy with Mitchell's homology of categories. Generalizing computations of McCarthy, we determine, for a discrete ring $R$, the $K$-theory of the exact category of finitely-generated projective $R$-modules with coefficients in the $n$-fold smash product functor. This computation allows us to analyze the effects of applying this functorial construction to the Goodwillie Taylor tower of a homotopy endofunctor of spectra. In the case of $\Sigma^\infty\Omega^\infty$, the associated tower recovers the Taylor tower of relative $K$-theory as computed by Lindenstrauss and McCarthy. Submission published under a 24 month embargo labeled 'Closed Access', the embargo will last until 2019-08-01 The student, Juan Villeta-Garcia, accepted the attached license on 2017-07-13 at 00:31. The student, Juan Villeta-Garcia, submitted this Dissertation for approval on 2017-07-13 at 00:39. DSpace SAF Submission Ingestion Package generated from Vireo submission #11445 on 2017-09-29 at 11:19:20 Made available in DSpace on 2017-09-29T17:52:21Z (GMT). No. of bitstreams: 3 VILLETA-GARCIA-DISSERTATION-2017.pdf: 750526 bytes, checksum: e448e5b89a1da1f31d4da897cb9deef9 (MD5) LICENSE.txt: 4216 bytes, checksum: c38291b8b3814fb92c0b22eebd6c8abf (MD5) PROQUEST_LICENSE.txt: 4562 bytes, checksum: 7617e8857ed978c9e2e13bd6432a3559 (MD5) Previous issue date: 2017-07-13 Embargo set by: Colleen Fallaw for item 103498 Lift date: 2019-09-29T17:52:45Z Reason: Author requested closed access (OA after 2yrs) in Vireo ETD system Limited Restriction Lifted for Item 103498 on 2019-09-30T09:15:23Z. application/pdf http://hdl.handle.net/2142/98290 en Copyright 2017 Juan Villeta-Garcia Algebraic K-theory Goodwillie calculus Homotopy theory Algebraic topology Stabilizing spectral functors of exact categories text text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.