Related papers: A Full and faithful Nerve for 2-categories
Multimodal normal incestual systems are investigated in terms of multiple categories. The different sorted composition of operators are exhibited as 2-cells in multiple categories built up from 2-categories giving rise to different axioms.…
We develop a homotopical framework for small categories that extends classical invarints of algebraic topology to the categorical setting. Our approach is based on the construction of genuine path category, obtained trough a localization…
We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan…
We show that diagram groups can be viewed as fundamental groups of spaces of positive paths on directed 2-complexes (these spaces of paths turn out to be classifying spaces). Thus diagram groups are analogs of second homotopy groups,…
In this article we discuss local aspects of 2-functors defined on the path 2-groupoid of a smooth manifold; in particular, local trivializations and descent data. This is a contribution to a project that provides an axiomatic formulation of…
Given a locally finite cover of a simplicial complex by subcomplexes, Bj\"orner's version of the Nerve Theorem provides conditions under which the homotopy groups of the nerve agree with those of the original complex through a range of…
We give a simple proof that complete Segal animae are equivalent to categories.
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…
For every adjunction of stable $\infty$-categories -- or more generally, in any locally stable $(\infty,2)$-category -- we give a simple procedure for inverting the twist and cotwist functors associated to this adjunction. As a consequence,…
This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen's Theorem B and Thomason's Homotopy Colimit…
In homotopy theory, exact sequences and spectral sequences consist of groups and pointed sets, linked by actions. We prove that the theory of such exact and spectral sequences can be established in a categorical setting which is based on…
We give another proof of the fact that there is a dual equivalence between the $\infty$-category of monoidal $\infty$-categories with left adjoint oplax monoidal functors and that with right adjoint lax monoidal functors by constructing a…
In this paper we call generalized lax epimorphism a functor defined on a ring with several objects, with values in an abelian AB5 category, for which the associated restriction functor is fully faithful. We characterize such a functor with…
There is a free construction from multicategories to permutative categories, left adjoint to the endomorphism multicategory construction. The main result shows that these functors induce an equivalence of homotopy theories. This result…
Consider the obvious functor from the unbounded derived category of all finitely generated modules over a left noetherian ring $R$ to the unbounded derived category of all modules. We answer the natural question whether this functor defines…
We formulate and prove a twofold generalisation of Lie's second theorem that integrates homomorphisms between formal group laws to homomorphisms between Lie groups. Firstly we generalise classical Lie theory by replacing groups with…
The goal of the present paper is to compare, in a precise way, two notions of operads up to homotopy which appear in the literature. Namely, we construct a functor from the category of strict unital homotopy colored operads to the category…
A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is…
We extend the homotopy theories based on point reduction for finite spaces and simplicial complexes to finite acyclic categories and $\Delta$-complexes, respectively. The functors of classifying spaces and face posets are compatible with…
There is a well-known correspondence between coherent theories (and their interpretations) and coherent categories (resp. functors), hence the (2,1)-category $\mathbf{Coh_{\sim}}$ (of small coherent categories, coherent functors and all…