Linear quasi-categories as templicial modules

We introduce a notion of enriched $\infty$-categories over a suitable monoidal category, in analogy with quasi-categories over the category of sets. We make use of certain colax monoidal functors, which we call templicial objects, as a variant of simplicial objects respecting the monoidal structure. We relate the resulting enriched quasi-categories to nonassociative Frobenius monoidal functors, allowing us to prove that the free templicial module over an ordinary quasi-category is a linear quasi-category. To any dg-category we associate a linear quasi-category, the linear dg-nerve, which enhances the classical dg-nerve. Finally, we prove an equivalence between (homologically) non-negatively graded dg-categories on the one hand and templicial modules with a Frobenius structure on the other hand, indicating that nonassociative Frobenius templicial modules and linear quasi-categories can be seen as relaxations of dg-categories.Comment: 68 pages, no figures; revised introduction, added references, revised section 4 for readability, results unchange

Convergence and quantale-enriched categories

Generalising Nachbin's theory of "topology and order", in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $\mathcal{V}$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category

On the tensor product of linear sites and Grothendieck categories

We define a tensor product of linear sites, and a resulting tensor product of Grothendieck categories based upon their representations as categories of linear sheaves. We show that our tensor product is a special case of the tensor product of locally presentable linear categories, and that the tensor product of locally coherent Grothendieck categories is locally coherent if and only if the Deligne tensor product of their abelian categories of finitely presented objects exists. We describe the tensor product of non-commutative projective schemes in terms of Z-algebras, and show that for projective schemes our tensor product corresponds to the usual product scheme.Comment: New sections 5.3 on the alpha-Deligne tensor product and 5.4 on future prospect