Related papers: A calculus on lax functors

We generalize Quillen's Theorem A to diagrams of lax 2-functors which commute up to transformation. It follows from a special case of this result that 2-categories are models for homotopy types.

In this paper, we state and prove precise theorems on the classification of the category of (braided) categorical groups and their (braided) monoidal functors, and some applications obtained from the basic studies on monoidal functors…

Monoidal categories enriched in a braided monoidal category $\mathcal{V}$ are classified by braided oplax monoidal functors from $\mathcal{V}$ to the Drinfeld centers of ordinary monoidal categories. In this article, we prove that this…

For an arbitrary symmetric monoidal $\infty$-category $\mathcal{V}$, we define the factorization homology of $\mathcal{V}$-enriched $(\infty,1)$-categories over (possibly stratified) 1-manifolds and study some of its basic properties. In…

Thomason's Homotopy Colimit Theorem has been extended to bicategories and this extension can be adapted, through the delooping principle, to a corresponding theorem for diagrams of monoidal categories. In this version, we show that the…

Cofibration categories are a formalization of homotopy theory useful for dealing with homotopy colimits that exist on the level of models as colimits of cofibrant diagrams. In this paper, we deal with their enriched version. Our main result…

Let $\mathcal C$ be a category with finite colimits, and let $(\mathcal E,\mathcal M)$ be a factorisation system on $\mathcal C$ with $\mathcal M$ stable under pushouts. Writing $\mathcal C;\mathcal M^{\mathrm{op}}$ for the symmetric…

Let $\mathcal{B}$ be a subcategory of a given category $\mathcal{D}$. Let $\mathcal{B}$ has monoidal structure. In this article, we discuss when can one extend the monoidal structure of $\mathcal{B}$ to $\mathcal{D}$ such that $\mathcal{B}$…

This paper introduces lax orthogonal algebraic weak factorisation systems on 2-categories and describes a method of constructing them. This method rests in the notion of simple 2-monad, that is a generalisation of the simple reflections…

We consider a closed symmetric monoidal category $\mathcal{M}$. We show that if $I$ is a small category then $\mathcal{M}^I$ is a closed $\mathcal{M}$-module. We rewrite the Yoneda Lemma in the case of monoidal valued functors. We derive an…

We define Euler characteristic of a category enriched by a monoidal model category. If a monoidal model category V is equipped with Euler characteristic that is compatible with weak equivalences and fibrations in V, then our Euler…

Let $V$ be a finite set. Let $\mathcal{K}$ be a simplicial complex with its vertices in $V$. In this paper, we discuss some differential calculus on $V$. We construct some constrained homology groups of $\mathcal{K}$ by using the…

We introduce a notion of (co)presheaf on a lax double functor $X$, which we generally call an instance. In the terminology of double-categorical logic, a lax double functor valued in sets, possibly preserving finite products, is called a…

We give an account of lax orthogonal factorisation systems on order-enriched categories. Among them, we define and characterise the KZ-reflective ones, in a way that mirrors the characterisation of reflective orthogonal factorisation…

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to…

We introduce a notion of "weak model category" which is a weakening of the notion of Quillen model category, still sufficient to define a homotopy category, Quillen adjunctions, Quillen equivalences and most of the usual construction of…

We provide a multiplicative classification of polynomial endofunctors on spectra in terms of their Mackey functors of cross--effects. More precisely, we prove that various categories of multivariable excisive functors from spectra to…

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal n-excisive approximation, which may be thought of as its n-excisive part. Homogeneous functors,…

We establish a formal correspondence between resource calculi an appropriate linear multicategories. We consider the cases of (symmetric) representable, symmetric closed and autonomous multicategories. For all these structures, we prove…

Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its…