Related papers: Exact completion and constructive theories of sets
We explain why the naive definition of a natural exact category structure on complete, separated topological vector spaces with linear topology fails. In particular, contrary to arXiv:0711.2527, the category of such topological vector…
We develop techniques for constructing model structures on chain complexes valued in accessible exact categories, and apply this to show that for a closed symmetric monoidal, locally presentable exact category $\mathpzc{E}$ with exact…
In an abelian category $\mathscr{A}$ with small ${\rm Ext}$ groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors $\mathcal{F}$ of ${\rm Ext}^{1}_{\mathscr{A}}(-,-)$ such…
We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple…
We find the model completion of the theory modules over $A$, where $A$ is a finitely generated commutative algebra over a field $K$. This is done in a context where the field $K$ and the module are represented by sorts in the theory, so…
We construct the reduction of an exact category with a twist functor with respect to an element of its graded center in presence of an exact-conservative forgetful functor annihilating this central element. The procedure allows, e.g., to…
Matrix completion results deal with the question of when a partially specified symmetric matrix can be completed to a member of certain matrix cones. Results from positive semidefinite matrix completion and completely positive matrix…
A certain amount of category theory is developed in an arbitrary finitely complete category with a factorization system on it, playing the role of the comprehensive factorization system on Cat. Those aspects related to the concepts of…
For an exact category we provide two constructions of an ambient category in which the initial category is resolving: In the derived category and in the Gabriel--Quillen embedding. For the first construction we describe a pre-aisle and its…
We argue that locally Cartesian closed categories form a suitable doctrine for defining dependent type theories, including non-extensional ones. Using the theory of sketches, one may define syntactic categories for type theories in a style…
For a (possibly large) realized limit sketch $\mathcal{S}$ such that every $\mathcal{S}$-model is small in a suitable sense we show that the category of cocontinuous functors $\mathsf{Mod}(\mathcal{S}) \to \mathcal{C}$ into a cocomplete…
We study the category pro-SSet of pro-simplicial sets, which arises in etale homotopy theory, shape theory, and pro-finite completion. We establish a model structure on pro-SSet so that it is possible to do homotopy theory in this category.…
The development of category theory in univalent foundations and the formalization thereof is an active field of research. Categories in that setting are often assumed to be univalent which means that identities and isomorphisms of objects…
Several important types of categories have been shown to be both exact and coexact (in the sense of Barr). The first type consists of abelian categories, which due to their self-dual definition, can be seen to be both exact and coexact by…
The focus of this article is on metric completions of triangulated categories arising in the representation theory of hereditary finite dimensional algebras and commutative rings. We explicitly describe all completions of bounded derived…
With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions…
For a relational Horn theory $\mathbb{T}$, we provide useful sufficient conditions for the exponentiability of objects and morphisms in the category $\mathbb{T}\text{-}\mathsf{Mod}$ of $\mathbb{T}$-models; well-known examples of such…
We initiate a systematic study of the perfection of affine group schemes of finite type over fields of positive characteristic. The main result intrinsically characterises and classifies the perfections of reductive groups, and obtains a…
We contribute to the knowledge of the quantifier completions and their applications by using the language of doctrines. This algebraic presentation allows us to properly analyse the behaviour of the existential and universal quantifiers. We…
We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits and jointly strongly epimorphic pairs of arrows. We give…