Related papers: Yoneda Lemma for $\mathcal{D}$-Simplicial Spaces

We give the definitions of model bicategory and $q$-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory $\mathcal{C}$, denote by…

In [Joyal] where the category $\Theta$ is first defined it is noted that the dimensional shift on $\Theta$ suggests an elegant presentation of the unreduced suspension on cellular sets. In this note we prove that the reduced suspension…

We generalise the usual notion of fibred category; first to fibred 2-categories and then to fibred bicategories. Fibred 2-categories correspond to 2-functors from a 2-category into 2-Cat. Fibred bicategories correspond to trihomomorphisms…

We formulate the concept of minimal fibration in the context of fibrations in the model category $\mathbf{S}^\mathcal{C}$ of $\mathcal{C}$-diagrams of simplicial sets, for a small index category $\mathcal{C}$. When $\mathcal{C}$ is an…

The description of algebraic structure of n-fold loop spaces can be done either using the formalism of topological operads, or using variations of Segal's $\Gamma$-spaces. The formalism of topological operads generalises well to different…

We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of…

We utilise the theory of crossed simplicial groups to introduce a collection of local Quillen model structures on the category of simplicial presheaves with a compact planar Lie group action on a small Grothendieck site. As an application,…

We prove a correspondence between $\kappa$-small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively $\kappa$-compact maps in their…

Yoneda'e Lemma is about the canonical isomorphism of all the natural transformations from a given representable covariant (contravariant, reps.) functor (from a locally small category to the category of sets) to a covariant (contravariant,…

Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op} \rightsquigarrow {\bf Cat}$, while the special case of discrete fibrations encode presheaves $B^{op} \to {\bf Set}$. A two-sided…

We observe that the notion of a trivial Serre fibration, a Serre fibration, and being contractible, for finite CW complexes, can be defined in terms of the Quillen lifting property with respect to a single map M-->/\ of finite topological…

We introduce and develop the notion of *displayed categories*. A displayed category over a category C is equivalent to "a category D and functor F : D --> C", but instead of having a single collection of "objects of D" with a map to the…

We introduce the notion of an effective Kan fibration, a new mathematical structure that can be used to study simplicial homotopy theory. Our main motivation is to make simplicial homotopy theory suitable for homotopy type theory. Effective…

If $D$ is a Reedy category and $M$ is a model category, the category $M^{D}$ of $D$-diagrams in $M$ is a model category under the Reedy model category structure. If $C \to D$ is a Reedy functor between Reedy categories, then there is an…

We show that there is a model structure in the sense of Quillen on an arbitrary Frobenius category $\F$ such that the homotopy category of this model structure is equivalent to the stable category $\underline{\F}$ as triangulated…

We study the covariant model structure on dendroidal spaces and establish direct relations to the homotopy theory of algebras over a simplicial operad as well as to the homotopy theory of special $\Gamma$-spaces. As an important tool in the…

Lie $\infty$-groupoids are simplicial Banach manifolds that satisfy an analog of the Kan condition for simplicial sets. An explicit construction of Henriques produces certain Lie $\infty$-groupoids called `Lie $\infty$-groups' by…

A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and…

We give a new construction of the model structure on the category of simplicial sets for homotopy $n$-types, originally due to Elvira-Donazar and Hernandez-Paricio, using a right transfer along the coskeleton functor. We observe that an…

Grothendieck fibrations are fundamental in capturing the concept of dependency, notably in categorical semantics of type theory and programming languages. A relevant instance are Dialectica fibrations which generalise G\"odel's Dialectica…