Barwick

Papers

12. Multiplicative structures on algebraic K-theory

• Submitted for publication (13 pages).
• arXiv:1304.4867
• Abstract. We prove that the algebraic K-theory is lax symmetric monoidal in a canonical fashion, and that this lax symmetric monoidal structure enjoys a universal property very similar to the one enjoyed by the algebraic K-theory functor itself.

11. From operator categories to topological operads.

• Submitted for publication (49 pages).
• arXiv:1302.5756
• Abstract. In this paper it is shown that an assortment of operads near to many topologists' hearts enjoy (homotopy) universal properties of an expressly combinatorial nature. These include the operads A_n and E_n. The main idea that makes this possible is the notion of an operator category, which controls the homotopy types of these operads in a strong sense.

10. On the Q construction for exact infinity-categories

• Preprint (9 pages).
• arXiv:1301.4725
• Abstract. We prove that the K-theory of an exact quasicategory can be computed via a higher categorical variant of the Q construction.

9. On exact infinity-categories and the Theorem of the Heart

• Submitted for publication (19 pages).
• arXiv:1212.5232
• Abstract. We introduce a notion of exact quasicategory, and we prove an analogue of Amnon Neeman's Theorem of the Heart for Waldhausen K-theory.

8. Quillen theorems Bn for homotopy pullbacks of (infinity, k)- categories. (with Dan Kan)

• Submitted for publication (28 pages).
• arXiv:1101.4879
• arXiv:1208.1777
• Abstract. We extend the Quillen Theorem Bn for homotopy fibers of Dwyer, et al. to similar results for homotopy pullbacks and note that these results imply similar results for zigzags in the categories of relative categories and k-relative categories, not only with respect to their Reedy structures but also their Rezk structure, which turns them into models for the theories of (∞, 1)- and (∞, k)-categories, respectively.

7. On the algebraic K-theory of higher categories.

• Submitted for publication (99 pages).
• arXiv:1204.3607
• Abstract. We prove that Waldhausen K-theory, when extended to a very general class of quasicategories, can be described as a Goodwillie differential. In particular, K-theory spaces admit canonical (connective) deloopings, and the K-theory functor enjoys a universal property. Using this, we give new, higher categorical proofs of the Approximation, Additivity, and Fibration Theorems of Waldhausen. As applications of this technology, we study the algebraic K-theory of associative rings in various homotopical contexts and the algebraic K-theory of spectral Deligne-Mumford stacks.

6. On the unicity of the homotopy theory of higher categories. (with Chris Schommer-Pries)

• Submitted for publication (41 pages).
• arXiv:1112.0040
• Abstract. We propose four axioms that a quasicategory should satisfy to be considered a reasonable homotopy theory of (∞, n)-categories. This axiomatization requires that a homotopy theory of (∞, n)-categories, when equipped with a small amount of extra structure, satisfies a simple, yet surprising, universal property. We further prove that the space of such quasicategories is homotopy equivalent to the n-th power of B(Z/2). In particular, any two such quasicategories are equivalent. This generalizes a theorem of Toën when n = 1, and it verifies two conjectures of Simpson. We also provide a large class of examples of models satisfying our axioms, including those of Joyal, Kan, Lurie, Rezk, and Simpson.

5. From partial model categories to infinity-categories. (with Dan Kan)

• Submitted for publication (17 pages).
• arXiv:1102.2512
• arXiv:1101.4879
• Abstract. In this article we study the problem of extracting an ∞-category from a relative category. We introduce partial model categories, which are relative categories that satisfy mild versions of the axioms of a model category. Since these axioms involve only the weak equivalences, they are general enough to include the vast majority of the relative categories one encounters in practice. We show that the simplicial nerve of a partial model category is "essentially" a complete Segal space, generalizing a result of Charles Rezk. To prove this, we must introduce a significant generalization of a Quillen's Theorem B. We show also that, conversely, any complete Segal space is dimensionwise equivalent to the simplicial nerve of a partial model category.

4. n-relative categories: A model for the homotopy theory of n-fold homotopy theories. (with Dan Kan)

• To appear in Homology, Homotopy, and Applications (18 pages).
• arXiv:1102.0186
• Abstract. We introduce, for every positive integer n, the notion of an n-relative category and show that the category of the small n-relative categories is a model for the homotopy theory of n-fold homotopy theories, i.e. homotopy theories of … of homotopy theories.

3. A characterization of simplicial localization functors and a discussion of DK equivalences. (with Dan Kan)

• Indagationes Mathematicae 23 (2012), pp. 69-79. DOI: 10.1016/j.indag.2011.10.001
• arXiv:1012.1540
• arXiv:1012.1541
• Abstract. In a previous paper, we lifted Charles Rezk’s complete Segal model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories. Here, we characterize simplicial localization functors among relative functors from relative categories to simplicial categories as any choice of homotopy inverse to the delocalization functor of Dwyer and the second author. We employ this characterization to obtain a more explicit description of the weak equivalences in the model category of relative categories mentioned above by showing that these weak equivalences are exactly the DK-equivalences, i.e. those maps between relative categories which induce a weak equivalence between their simplicial localizations.

2. Relative categories: Another model for the homotopy theory of homotopy theories. (with Dan Kan)

• Indagationes Mathematicae 23 (2012), pp. 42-68. DOI: 10.1016/j.indag.2011.10.002
• arXiv:1101.0772
• arXiv:1011.1691
• Abstract. We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.

1. On left and right model categories and left and right Bousfield localizations.

• Homology, Homotopy and Applications, Vol. 12 (2010), No. 2, pp. 245-320. HHA152.
• arXiv:0708.2067
• arXiv:0708.2832
• arXiv:0708.3435
• Abstract. We verify the existence of left Bousfield localizations and of enriched left Bousfield localizations, and we prove a collection of useful technical results characterizing certain fibrations of (enriched) left Bousfield localizations. We also use such Bousfield localizations to construct a number of new model categories, including models for the homotopy limit of right Quillen presheaves, for Postnikov towers in model categories, and for presheaves valued in a symmetric monoidal model category satisfying a homotopy-coherent descent condition. We then verify the existence of right Bousfield localizations of right model categories, and we apply this to construct a model of the homotopy limit of a left Quillen presheaf as a right model category.