arXiv subjects
Publications and source records attributed to Fernando Muro.
We correct a mistake in the construction of push-outs along free morphisms of algebras over a nonsymmetric operad in arXiv:1101.1634 [math.AT], and we fix the affected results in arXiv:1101.1634 [math.AT] and arXiv:1304.6641 [math.AT].
We construct small cylinders for cellular non-symmetric DG-operads over an arbitrary commutative ring by using the basic perturbation lemma from homological algebra. We show that our construction, applied to the A-infinity operad, yields the operad parametrizing A-infinity maps whose linear part is the identity. We also compute some other examples with non-trivial operations in arities 1 and 0.
In this paper we show how to modify cofibrations in a monoidal model category so that the tensor unit becomes cofibrant while keeping the same weak equivalences. We obtain aplications to enriched categories and coloured operads in stable homotopy theory.
We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures.
We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors which induce weak equivalences on morphism objects and equivalences of ordinary categories when we take sets of connected components on morphism objects.
In this paper we study spaces of algebras over an operad (non-symmetric) in symmetric monoidal model categories. We first compute the homotopy fiber of the forgetful functor sending an algebra to its underlying object, extending a result of Rezk. We then apply this computation to the construction of geometric moduli stacks of algebras over an operad in a homotopical algebraic geometry context in the sense of Toën and Vezzosi. We show under mild hypotheses that the moduli stack of unital associative algebras is a Zariski open substack of the moduli stack of non-necessarily unital associative algebras. The classical analogue for finite-dimensional vector spaces was noticed by Gabriel.
We prove, under mild assumptions, that a Quillen equivalence between symmetric monoidal model categories gives rise to a Quillen equivalence between their model categories of (non-symmetric) operads, and also between model categories of algebras over operads. We also show left properness results on model categories of operads and algebras over operads. As an application, we prove homotopy invariance for (unital) associative operads.
In this paper we introduce a new approach to determinant functors which allows us to extend Deligne's determinant functors for exact categories to Waldhausen categories, (strongly) triangulated categories, and derivators. We construct universal determinant functors in all cases by original methods which are interesting even for the known cases. Moreover, we show that the target of each universal determinant functor computes the corresponding $K$-theory in dimensions 0 and 1. As applications, we answer open questions by Maltsiniotis and Neeman on the $K$-theory of (strongly) triangulated categories and a question of Grothendieck to Knudsen on determinant functors. We also prove additivity theorems for low-dimensional $K$-theory and obtain generators and (some) relations for various $K_{1}$-groups.
We construct a topological cellular operad such that the algebras over its cellular chains are the homotopy unital A-infinity algebras of Fukaya-Oh-Ohta-Ono.
In this paper we show an example of two differential graded algebras that have the same derivator K-theory but non-isomorphic Waldhausen K-theory. We also prove that Maltsiniotis's comparison and localization conjectures for derivator K-theory cannot be simultaneously true.
We endow categories of non-symmetric operads with natural model structures. We work with no restriction on our operads and only assume the usual hypotheses for model categories with a symmetric monoidal structure. We also study categories of algebras over these operads in enriched non-symmetric monoidal model categories.
We show that K_1 of an exact category agrees with K_1 of the associated triangulated derivator. More generally we show that K_1 of a Waldhausen category with cylinders and a saturated class of weak equivalences coincides with K_1 of the associated right pointed derivator.
We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories.
We exhibit a triangulated category which is neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category.
The primary algebraic model of a ring spectrum is the ring of homotopy groups. We introduce the secondary model which has the structure of a secondary analogue of a ring. This new algebraic model determines Massey products and cup-one squares. As an application we obtain new derivations of the homotopy ring.
We give a small functorial algebraic model for the 2-stage Postnikov section of the K-theory spectrum of a Waldhausen category and use our presentation to describe the multiplicative structure with respect to biexact functors.
In this paper we prove the laws of Toda brackets on the homotopy groups of a connective ring spectrum and the laws of the cup-one square in the homotopy groups of a commutative connective ring spectrum.
We give a simple representation of all elements in K_1 of a Waldhausen category and prove relations between these representatives which hold in K_1.