Related papers: Model $\infty$-categories III: the fundamental the…

This note extends Quillen's Theorem A to a large class of categories internal to topological spaces. This allows us to show that under a mild condition a fully faithful and essentially surjective functor between such topological categories…

Categories enriched in the opposite poset of non-negative reals can be viewed as generalizations of metric spaces, known as Lawvere metric spaces. In this article, we develop model structures on the categories…

We define and study the $(\infty,2)$-category $\mathbf{Cat}_{\infty}(\mathcal{C})$ of $(\infty,1)$-categories internal to a general $(\infty,1)$-category $\mathcal{C}$ via an associated externalization construction. In the first part, we…

Our aim is to construct fibrewise localizations in model categories. For pointed spaces, the general idea is to decompose the total space of a fibration as a diagram over the category of simplices of the base and replace it by the localized…

Using the theory of extensions of L-infinity algebras, we construct rational homotopy models for classifying spaces of fibrations, giving answers in terms of classical homological functors, namely the Chevalley-Eilenberg and Harrison…

We show that C if is a proper model category, then the pro-category pro-C has a strict model structure in which the weak equivalences are the levelwise weak equivalences. The strict model structure is the starting point for many homotopy…

We show that the category of N-complexes has a Str\om model structure, meaning the weak equivalences are the chain homotopy equivalences. This generalizes the analogous result for the category of chain complexes (N = 2). The trivial objects…

In a recent paper we introduced a much weaker and easy to verify structure than a model category, which we called a "weak fibration category". We further showed that a small weak fibration category can be "completed" into a full model…

An appropriate framework is put forward for the construction of $\lambda$-models with $\infty$-groupoid structure, which we call \textit{homotopic $\lambda$-models}, through the use of an $\infty$-category with cartesian closure and enough…

Let $G$ be a discrete group. We prove that the category of $G$-posets admits a model structure that is Quillen equivalent to the standard model structure on $G$-spaces. As is already true nonequivariantly, the three classes of maps defining…

We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is…

We give a criterion on a group $\pi$ and a homomorphism $w \colon \pi \to C_2$ under which closed $4$-manifolds with fundamental group $\pi$ and orientation character $w$ are classified up to homotopy equivalence by their quadratic…

We show that a category $\mathscr{M}$ equipped with a model structure defined by a proper, locally small class of orbits $\mathscr{O}$ is Quillen equivalent to the category of small relative presheaves…

For each manifold or effective orbifold $Y$ and commutative ring $R$, we define a new homology theory $MH_*(Y;R)$, $M$-$homology$, and a new cohomology theory $MH^*(Y;R)$, $M$-$cohomology$. For $MH_*(Y;R)$ the chain complex…

Given a map $f: X\rightarrow Y$ of simply connected spaces of finite type such. The space of based loops at $f$ of the space of maps between $X$ and $Y$ is denoted by $\Omega_{f} Map(X,Y)$. For $n> 0$, we give a model categorical…

Algebraic topological methods have been used successfully in concurrency theory, the domain of theoretical computer science that deals with distributed computing. L. Fajstrup, E. Goubault, and M. Raussen have introduced partially ordered…

We construct a rational homotopy-theoretic model for a classifying space of locally conformally symplectic structures on four-manifolds, and use it to definition a cobordism category of three-manifolds `anchored' by principal $\Omega^2 S^2$…

Suppose that $F: \mathcal{N} \to \mathcal{M}$ is a functor whose target is a Quillen model category. We give a succinct sufficient condition for the existence of the right-induced model category structure on $\mathcal{N}$ in the case when…

In the rational cohomology of a 1-connected space a structure of $C_{\infty}$-algebra is constructed and it is shown that this object determines the rational homotopy type

We prove the first equivalence between a weak non-algebraic model and a semi-strict algebraic model of $(\infty, n)$-categories. This takes the form of a natural semi-strictification, whereby a weak $(\infty, n)$-category is embedded into a…