Related papers: A convenient category of cubes

The object of this paper is to prove that the standard categories in which homotopy theory is done, such as topological spaces, simplicial sets, chain complexes of abelian groups, and any of the various good models for spectra, are all…

We present a way of constructing a Quillen model structure on a full subcategory of an elementary topos, starting with an interval object with connections and a certain dominance. The advantage of this method is that it does not require the…

A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets…

In this paper we investigate important categories lying strictly between the Kleisli category and the Eilenberg-Moore category, for a Kock-Z\"oberlein monad on an order-enriched category. Firstly, we give a characterisation of free algebras…

We prove that an \'etale fibration between $L_\infty$-bundles admits local sections composed of several elementary morphisms of particularly simple and accessible type. As applications, we establish an inverse function theorem for…

We develop a Gabriel-Morita theory for strong monads on pointed monoidal model categories. Assuming that the model category is excisive, i.e. the derived suspension functor is conservative, we show that if the monad T preserves cofibre…

We study properties of the cubical Joyal model structures on cubical sets by means of a combinatorial construction which allows for convenient comparisons between categories of cubical sets with and without symmetries. In particular, we…

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit S forms a commutative ring. An idempotent e of this ring will split the homotopy category. We prove that provided the…

We consider representations of quivers taking values in monads or comonads over a Grothendieck category $\mathcal C$. We treat these as scheme like objects whose ``structure sheaf'' consists of monads or comonads. By using systems of…

We construct Quillen equivalences between the model categories of monoids (rings), modules and algebras over two Quillen equivalent model categories under certain conditions. This is a continuation of our earlier work where we established…

We construct a monoidal version of Lurie's un/straightening equivalence. In more detail, for any symmetric monoidal $\infty$-category $\mathbf C$, we endow the $\infty$-category of coCartesian fibrations over $\mathbf C$ with a (naturally…

We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups $S$ and $T$, there is another Cuntz semigroup $[[S,T]]$ playing the role of morphisms from $S$ to $T$. Applied to C$^*$-algebras…

We prove that the category of vector bundles over a fixed smooth manifold and its corresponding category of convenient modules are models for intuitionistic differential linear logic. The exponential modality is modelled by composing the…

We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category $\mathcal{C}$. For the discrete case, we build a model structure on the slice…

The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring $B$ can be embedded as a right coideal subalgebra into a Hopf algebra $A$ such that $A$ is faithfully flat as a $B$-module. In…

This thesis provides an introduction to the various category theory ideas employed in topological quantum field theory. These theories are viewed as symmetric monoidal functors from topological cobordism categories into the category of…

Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasi-category admitting…

Quillen showed that simplicial sets form a model category (with appropriate choices of three classes of morphisms), which organized the homotopy theory of simplicial sets. His proof is very difficult and uses even the classification theory…

In categories of linear relations between finite dimensional vector spaces, composition is well-behaved only at pairs of relations satisfying transversality and monicity conditions. A construction of Wehrheim and Woodward makes it possible…

We study structures which have arisen in recent work by the present author and Bob Coecke on a categorical axiomatics for Quantum Mechanics; in particular, the notion of strongly compact closed category. We explain how these structures…