Simplicial Nerve of an Ainf category

2015

In this paper we define a sequence of monads T(∞,n)(n ∈ N) on the category ∞-Gr of ∞-graphs. We conjecture that algebras for T(∞,0), which are defined in a purely algebraic setting, are models of∞-groupoids. More generally, we conjecture that T(∞,n)-algebras are models for (∞, n)-categories. We prove that our (∞, 0)-categories are bigroupoids when truncated at level 2. Introduction The notion of weak (∞, n)-category can be made precise in many ways depending on our approach to higher categories. Intuitively this is a weak∞-category such that all its cells of dimension greater than n are equivalences. Models of weak (∞, 1)-categories (case n = 1) are diverse: for example there are the quasicategories studied by Joyal and Tierney (see [24]), but also there are other models which have been studied like the Segal categories, the complete Segal spaces, the simplicial categories, the topological categories, the relative categories, and there are known to be equivalent (a survey of models ...

Homology, Homotopy and Applications, 2007

The model structure on the category of chain functors Ch, developed in [4], has the main features of a simplicial model category structure, taking into account the lack of arbitrary (co-)limits in Ch. After an appropriate tensor and cotensor structure in Ch is established (§1, §3), Quillen's axiom SM7 is verified in §5 and §6. Moreover, it turns out that in the definition of a simplicial model structure, the category of simplicial sets can be replaced by the category of simplicial spectra endowing Ch with the structure of an approximate simplicial stable model structure (= approximate ss-model structure) (§7). In §8 the model structure on Ch is shown to be proper.

arXiv (Cornell University), 2020

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit a convenient notion of a mapping space, which we use to characterize the weak equivalences between fibrant objects in our model structure as DK-equivalences. This statement has an immediate corollary, which we will apply in practice: Corollary 1.12. Let F : C → D be a left adjoint between model categories and suppose that fibrations between fibrant objects in C are characterized by right lifting against a class S. If F preserves cofibrations and sends S to trivial cofibrations, then F is a left Quillen functor. Proposition 1.13 ([Hov99, Cor. 1.3.16]). Let F : C ⇄ D : U be a Quillen adjunction between model categories. Then the following are equivalent. (i) F ⊣ U is a Quillen equivalence. (ii) F reflects weak equivalences between cofibrant objects and, for every fibrant Y , the derived counit F U Y → Y is a weak equivalence. (iii) U reflects weak equivalences between fibrant objects and, for every cofibrant X, the derived unit X → U (F X) ′ is a weak equivalence.

Journal of Topology, 2020

In this paper we complete a chain of explicit Quillen equivalences between the model category for Θ n+1-spaces and the model category of small categories enriched in Θn-spaces. The Quillen equivalences given here connect Segal category objects in Θn-spaces, complete Segal objects in Θn-spaces, and Θ n+1-spaces.

We develop a theory of curved A ∞-categories around equivalences of their module categories. This allows for a uniform treatment of curved and uncurved A ∞-categories which generalizes the classical theory of uncurved A ∞-algebras. Furthermore, the theory is sufficiently general to treat both Fukaya categories and categories of matrix factorizations, as well as to provide a context in which unitification and categorification of pre-categories can be carried out. Our theory is built around two functors: the adjoint algebra functor U e and the functor Q *. The bulk of the paper is dedicated to proving crucial adjunction and homotopy theorems about these functors. In addition, we explore the non-vanishing of the module categories and give a precise statement and proof the folk result known as "Positselski-Kontsevich vanishing".

Advances in Mathematics, 2002

We realise Joyal's cell category Y as a dense subcategory of the category of ocategories. The associated cellular nerve of an o-category extends the well-known simplicial nerve of a small category. Cellular sets (like simplicial sets) carry a closed model structure in Quillen's sense with weak equivalences induced by a geometric realisation functor. More generally, there exists a dense subcategory Y A of the category of % A-algebras for each o-operad A in Batanin's sense. Whenever A is contractible, the resulting homotopy category of % A-algebras (i.e. weak o-categories) is equivalent to the homotopy category of compactly generated spaces.

2020

We construct a model structure on the category of cubical sets with connections whose cofibrations are the monomorphisms and whose fibrant objects are defined by the right lifting property with respect to inner open boxes, the cubical analogue of inner horns. We show that this model structure is Quillen equivalent to the Joyal model structure on simplicial sets via the triangulation functor. As an application, we show that cubical quasicategories admit an elegant and canonical notion of a mapping space between two objects.

2013

This introduction to higher category theory is intended to a give the reader an intuition for what $(\infty,1)$-categories are, when they are an appropriate tool, how they fit into the landscape of higher category, how concepts from ordinary category theory generalize to this new setting, and what uses people have put the theory to. It is a rough guide to a vast terrain, focuses on ideas and motivation, omits almost all proofs and technical details, and provides many references.

Geometry & Topology, 2013

While many different models for .1; 1/-categories are currently being used, it is known that they are Quillen equivalent to one another. Several higher-order analogues of them are being developed as models for .1; n/-categories. In this paper, we establish model structures for some naturally arising categories of objects which should be thought of as .1; n/-categories. Furthermore, we establish Quillen equivalences between them.

American Journal of Mathematics, 2001

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or 'continuous', functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.