Sheaves of ordered spaces and interval theories

We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. A precise characterisation of this subcategory is provided. The ambient topos makes available some general homotopical machinery

Completions of Implicative Assemblies

We continue our work on implicative assemblies by investigating under which circumstances a subset $M \subseteq \mathcal{S}$ of the separator gives rise to a full lex subcategory $\mathbf{Asm}_M$ of the quasitopos $\mathbf{Asm}_{\mathcal{A}}$ of all implicative assemblies such that ${\mathbf{Asm}_M}_{reg/lex} \simeq \mathbf{Asm}_{\mathcal{A}}$. We establish a characterisation. What is more, this characterisation is relevant to the study of $\mathbf{Asm}_M$'s ex-completion, which turns out to be a topos

A folk model structure on omega-cat

We establish a model structure on the category of strict omega-categories. The constructions leading to the model structure in question are expressed entirely within the scope of omega-categories, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while cofibrant objects are exactly the free ones. Our model structure transfers to n-categories along right-adjoints, for each n, thus recovering the known cases n = 1 and n = 2.Comment: 33 pages, expanded version of the original 17 pages synopsis, new sections adde

Witt rings of quadratically presentable fields

This paper introduces an approach to the axiomatic theory of quadratic forms based on presentable partially ordered sets, that is partially ordered sets subject to additional conditions which amount to a strong form of local presentability. It turns out that the classical notion of the Witt ring of symmetric bilinear forms over a field makes sense in the context of quadratically presentable fields, that is, fields equipped with a presentable partial order inequationaly compatible with the algebraic operations. In particular, Witt rings of symmetric bilinear forms over fields of arbitrary characteristics are isomorphic to Witt rings of suitably built quadratically presentable fields