Related papers: A Polynomial Construction of Nerves for Higher Cat…

We study several structure aspects of functor categories from a small additive category to a module category, in particular the category F(A,K) of functors from finitely generated free modules over a commutative ring A to vector spaces over…

We continue the program of structural differential geometry that begins with the notion of a tangent category, an axiomatization of structural aspects of the tangent functor on the category of smooth manifolds. In classical geometry, having…

Let $F$ be the category with the set of objects $\bf N$ and morphisms being the functions between the standard finite sets of the corresponding cardinalities. Let $Jf:F\rightarrow Sets$ be the obvious functor from this category to the…

Many homotopy-coherent algebraic structures can be described by Segal-type limit conditions determined by an "algebraic pattern", bywhich we mean an $\infty$-category equipped with a factorization system and a collection of "elementary"…

In this paper we construct an analogue of Lurie's "unstraightening" construction that we refer to as the "comprehension construction". Its input is a cocartesian fibration $p \colon E \to B$ between $\infty$-categories together with a third…

Via the adjunction $ - \boldsymbol{\cdot} 1 \dashv \mathcal V(1,-) \colon \mathsf{Span}(\mathcal V) \to \mathcal V \text{-} \mathsf{Mat} $ and a cartesian monad $ T $ on an extensive category $ \mathcal V $ with finite limits, we construct…

For most models of $(\infty,2)$-categories an embedding of the $\infty$-category of 2-categories into that of $(\infty,2)$-categories has been constructed in the form of a nerve construction of some flavor. We prove that all those nerve…

We establish an explicit comparison between two constructions in homotopy theory: the left adjoint of the homotopy coherent nerve functor, also known as the rigidification functor, and the Kan loop groupoid functor. This is achieved by…

If T is a commutative monad on a cartesian closed category, then there exists a natural T-bilinear pairing from T(X) times the space of T(1)-valued functions on X ("integration"), as well as a natural T-bilinear action on T(X) by the space…

A combinatorial neural code $\mathscr C\subseteq 2^{[n]}$ is convex if it arises as the intersection pattern of convex open subsets of $\mathbb R^d$. We relate the emerging theory of convex neural codes to the established theory of oriented…

In earlier work, Batanin has shown that an important class of definitions of higher categories could be apprehended together simply as monads over globular sets. This allowed him to generalize the notion of polygraph, initially introduced…

For functors $L:\A\to \B$ and $R:\B\to \A$ between any categories $\A$ and $\B$, a {\em pairing} is defined by maps, natural in $A\in \A$ and $B\in \B$, $$\xymatrix{\Mor_\B (L(A),B) \ar@<0.5ex>[r]^{\alpha} & \Mor_\A…

In this paper, we generalize the principle of the Long-Moody construction for representations of braid groups to other groups, such as mapping class groups of surfaces. Namely, we introduce endofunctors over a functor category that encodes…

The fundamental construction underlying descent theory, the lax descent category, comes with a functor that forgets the descent data. We prove that, in any $2$-category $\mathfrak{A} $ with lax descent objects, the forgetful morphisms…

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…

By building on the notions of internal projective and injective objects in a module category introduced by Douglas, Schommer-Pries, and Snyder, we extend the reconstruction theory for module categories of Etingof and Ostrik. More…

Using the braided version of Lawvere's algebraic theories and Mac Lane's PROPs, we introduce polynomial identities for arbitrary algebraic structures in a braided monoidal category C as well as their codimensions in the case when C is…

This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through *-autonomous monoidal categories and related structures.

The structure of the category of matroids and strong maps is investigated: it has coproducts and equalizers, but not products or coequalizers; there are functors from the categories of graphs and vector spaces, the latter being faithful;…

We consider the 3-category $2\mathfrak{C}at$ whose objects are 2-categories, 1-morphisms are lax functors, 2-morphisms are lax transformations and 3-morphisms are modifications. The aim is to show that it carries interesting…