A smorgasbord of typos corrected throught the text; also note multiple statements have been added which may have changed the numbering of some of them
Changed the "upcoming work with Christoph Winges" to linked the now available preprint
Added many more refs throughout the body of the text, notably in the intro
Improved section 1.2 to give more insight into how the thesis works; also included a clear dependency charts for the statements whose proof is not yet available.
Added an explicit description of the objects of THH*(CatEx) afeter Proposition 1.8 (StructureTheoremForTraceTheories)
Added a proof to Proposition 2.4 (KthCatExVerdierLiminary) and corrected the statement to talk about Verdier motives.
Added Proposition 2.6 to prove what are the dualizable objects of CatEx
Improved the presentation of Example 3.10 to show one computes the spectrum of endofunctors of the generator S.
Improved the proof that (Sp^fin id) is compact in Theorem 3.18 to remove the size issue dependency (label PresentableTCatEx)
Added Corollary 3.21 (LStrongMonoidal) to prove that the cotangent complex is strong monoidal (and therefore Lace canonically has a lax-monoidal structure)
Dwongraded 4.6 (AdjPresSOD) from a Theorem to a Proposition
Changed 4.7 to say additive instead of semi-additive
Added missing credit to Blumberg-Gepner-Tabuada's original proof for Proposition 4.12 (KtheoryCatExLaxMon)
Reworked the discussion after 4.12 and turned 4.13 (KlaceLaxMon) into a Proposition with a short proof.
Downgraded 5.5 (CyclicBarInCatEx) from a Proposition to a Remark, since the proof simply refered to later statements.
Added a sentence referencing section 8 to stress the fact that THH and laced THH are two (very) different functors
Fixed a claim in the proof of Lemma 5.18 (CycIsMadeOfTraceEq) saying a diagram was in Δ whereas it was only in Cat (this does not affect anything else).
Slightly reordered the proof of Theorem 5.21 (UniversalPropOfCyc) so that it tells you first what criterion it tries to check before checking it
From 5.25 and throughtout: Fred now always refers to the fiberwise-reduction of F and Fcyc for the fiberwise-reduction of a laced F. The latter is defined in a new definition, 6.12 (FcycDefinition)
Explained what is a differentiable category before 5.26
Improved a little the proof of Proposition 5.30 (THHInitialLaxMonoidal) to explain more carefully what happens in the localization.
Lengthened the description of fiberwise and non-fiberwise Goodwillie calculus in section 6, notably defining what a cross-effect is.
Fully rewrote the proof of Lemma 6.5 (IndPairViaLaxEq)
Clarified Remark 6.6 (NotTrueForIndLace)
Reworked the end of section 6.2 so that the main Theorem 6.13 (FcycIsCyclicInvariantIfFVerdier) is about general Verdier-localizing invariants and the statement about cyclic K-theory is a Corollary 6.14 (CyclicKtheoryCyclicInvariant)
Improved the presentation of the part in the sketch of proof of 6.18 (AbstractHigherTaylorTower) which appeals to invariance under cyclic permutations of the first derivative.
Fixed the statement of Lemma 6.22 which was incorrect (or at least, for which I do know of a proof), and modified the proof in accordance.
Fixed the statement of Proposition 6.24 which was incorrect (or at least, for which I do know of a proof), and modified the discussion afterwards in accordance.
After 7.4, improved the discussion of the epicyclic category and its relation to the cyclic category
Downgraded a number of Theorem to Propositions in section 7
Added after 7.25 a discussion of the explicit consequence, notably drawing a diagram of all the maps that play an important role in further sections.
Added a warning in section 10.1 about potential differences between spectrification and stabilisation
After Theorem 10.9, added a forward reference to section 11.3 where applications of this result can be found.
Improved the proof of Theorem 10.14 to be legible.
Improved the formulation of Proposition 10.24 to be clearer as to what functor commutes with which colimits.
After 11.4, improved the explanation of the relation with the result of Dundas-Goodwillie-McCarthy
Added a new Corollary 11.13 (KtheoryInvarianceForResolutionPropertyStacks) drawing an explicit consequence of the above theorem and of results of section 10
Corrected many typos throught the intro and chapters 2 to 5
Added a structure of the thesis subsection in the intro with a neat little dependency graph
Added references next to statements in the intro for the corresponding results in the body of the text
Clarified in the introduction why section 7 is done the way it is done, and referenced the attempt by Ayala, Mazel-Gee and Rozyenblum
Clarified the relationship between (genuine) TR and the limit of the Taylor tower of laced K-theory, including when the latter converges. In particular, this has created two new corollaries (and therefore changed many numbers in the later part of the introduction)
Added a discussion of Almkvist and Dotto-Krause-Nikolaus-Patchkoria and their investigations of the map from laced K-theory to TR
Corrected the definition of the trace of a dualizable object in section 2
Improved the proof of Lemma 3.3 (label TCatEXIsBimod).
Improved the presentation and clarified reliance on heart/weight structures of Example 3.10 (label SqZeroVSLace)
Added a proof that Lace preserves filtered colimits and clarified the end of the proof of Theorem 3.18 (label PresentableTCatEx)
Added a small discussion after 5.17 (label TestSampleTrLike) that the small indeterminacy in the definition of trace equivalences does not matter for trace-like functors