109 research outputs found
Weak Factorizations, Fractions and Homotopies
We show that the homotopy category can be assigned to any category equipped with a weak factorization system. A classical example of this construction is the stable category of modules. We discuss a connection with the open map approach to bisimulations proposed by Joyal, Nielsen and Winskel
Algebraic K-theory and abstract homotopy theory
We decompose the K-theory space of a Waldhausen category in terms of its
Dwyer-Kan simplicial localization. This leads to a criterion for functors to
induce equivalences of K-theory spectra that generalizes and explains many of
the criteria appearing in the literature. We show that under mild hypotheses, a
weakly exact functor that induces an equivalence of homotopy categories induces
an equivalence of K-theory spectra.Comment: Final versio
From fractions to complete Segal spaces
We show that the Rezk classification diagram of a relative category admitting
a homotopical version of the two-sided calculus of fractions is a Segal space
up to Reedy-fibrant replacement. This generalizes the result of Rezk and
Bergner on the classification diagram of a closed model category, as well as
the result of Barwick and Kan on the classification diagram of a partial model
category.Comment: 21 pages, LaTeX. Changes in v3: added some expository material,
following suggestions by anonymous referee. (N.B. numbering has changed.
The Kapustin-Li formula revisited
We provide a new perspective on the Kapustin-Li formula for the duality
pairing on the morphism complexes in the matrix factorization category of an
isolated hypersurface singularity. In our context, the formula arises as an
explicit description of a local duality isomorphism, obtained by using the
basic perturbation lemma and Grothendieck residues. The non-degeneracy of the
pairing becomes apparent in this setting. Further, we show that the pairing
lifts to a Calabi-Yau structure on the matrix factorization category. This
allows us to define topological quantum field theories with matrix
factorizations as boundary conditions.Comment: 28 pages, 3 figures, comments welcom
Internal Languages of Finitely Complete -categories
We prove that the homotopy theory of Joyal's tribes is equivalent to that of
fibration categories. As a consequence, we deduce a variant of the conjecture
asserting that Martin-L\"of Type Theory with dependent sums and intensional
identity types is the internal language of -categories with finite
limits.Comment: 41 pages, minor revision
The bar derived category of a curved dg algebra
Curved A-infinity algebras appear in nature as deformations of dg algebras.
We develop the basic theory of curved A-infinity algebras and, in particular,
curved dg algebras. We investigate their link with a suitable class of dg
coalgebras via the bar construction and produce Quillen model structures on
their module categories. We define the analogue of the relative derived
category for a curved dg algebra.Comment: 38 pages, with figures, corrected typos. To appear in the Journal of
Pure and Applied Algebr
Two Models for the Homotopy Theory of Cocomplete Homotopy Theories
We prove that the homotopy theory of cofibration categories is equivalent to
the homotopy theory of cocomplete quasicategories. This is achieved by
presenting both homotopy theories as fibration categories and constructing an
explicit equivalence between them
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