Related papers: Enriched quasi-categories and the templicial homot…

The framework of templicial vector spaces was put forth in arXiv:2302.02484v2 as a suitable generalization of simplicial sets in order to develop a theory of enriched quasi-categories, called quasi-categories in vector spaces. We construct…

In the case of $(\infty,1)$-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams of…

We set up a general theory of weak or homotopy-coherent enrichment in an arbitrary monoidal $\infty$-category $\mathcal{V}$. Our theory of enriched $\infty$-categories has many desirable properties; for instance, if the enriching…

We introduce, for \(\C\) a regular Cartesian Reedy category a model category whose fibrant objects are an analogue of quasicategories enriched in simplicial presheaves on \(C\). We then develop a coherent realization and nerve for this…

For any category ${\mathcal E}$ and monad $T$ thereon, we introduce the notion of $T$-simplicial object in ${\mathcal E}$. Any $T$-category in the sense of Burroni induces a $T$-simplicial object as its nerve. This nerve construction…

We introduce necklicial nerve functors from enriched categories to simplicial sets, which include Cordier's homotopy coherent, Lurie's differential graded and Le Grignou's cubical nerves. It is shown that every necklicial nerve can be…

Templicial objects were put forth in arXiv:2302.02484v2 to set up a suitable simplicial framework for enriched quasi-categories. Following Leinster, these objects feature certain comultiplications as a replacement for outer face maps in the…

We prove an adjoint functor theorem in the setting of categories enriched in a monoidal model category $\mathcal V$ admitting certain limits. When $\mathcal V$ is equipped with the trivial model structure this recaptures the enriched…

We follow the work of Aguiar on internal categories and introduce simplicial objects internal to a monoidal category as certain colax monoidal functors. Then we compare three approaches to equipping them with a discrete set of vertices. We…

We prove a rectification theorem for enriched infinity-categories: If V is a nice monoidal model category, we show that the homotopy theory of infinity-categories enriched in V is equivalent to the familiar homotopy theory of categories…

This paper introduces a skew variant of the notion of enriched category, suitable for enrichment over a skew-monoidal category, the main novelty of which is that the elements of the enriched hom-objects need not be in bijection with the…

The homotopy coherent nerve from simplicial categories to simplicial sets and its left adjoint C are important to the study of (infinity,1)-categories because they provide a means for comparing two models of their respective homotopy…

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model…

We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for…

We introduce the notion of a monoidal category enriched in a braided monoidal category $\mathcal V$. We set up the basic theory, and prove a classification result in terms of braided oplax monoidal functors to the Drinfeld center of some…

If $\mathscr{M}$ is a model category and $\mathcal{U}: \mathscr{A} \rightarrow \mathscr{M}$ is a functor, we defined a Quillen-Segal $\mathcal{U}$-object as a weak equivalence $\mathscr{F}: s(\mathscr{F}) \xrightarrow{\sim} t(\mathscr{F})$…

We define and study opfibrations of $V$-enriched categories when $V$ is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with…

We study the *homotopy theory* of $\infty$-categories enriched in the $\infty$-category $sS$ of simplicial spaces. That is, we consider $sS$-enriched $\infty$-categories as presentations of ordinary $\infty$-categories by means of a "local"…

Tensor products are ubiquitous in algebra, topology, logic and category theory. The present paper explores the monoidal structure of the category $\mathcal{V}\hspace{0pt}\mbox{-}\hspace{.5pt}\mathbf{Sup}$ of separated cocomplete enriched…

Many monoidal-type objects are known to be classified by maps from the Catalan simplicial set $\mathbb{C}$ to various nerves of categories and higher categories. There are, for example, three different nerves of the 2-category of categories…