Using techniques due to Dwyer–Greenlees–Iyengar we construct weight structures in triangulated categories generated by compact objects. We apply our result to show that, for a dg category whose homology vanishes in negative degrees and is semi-simple in degree 0, each simple module over the homology lifts to a dg module which is unique up to isomorphism in the derived category. This allows us, ...

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about parti...

Let Σ be a ranked set. A categorical Σ-algebra, cΣa for short, is a small category C equipped with a functor σC : C n //C, for each σ ∈ Σn, n ≥ 0. A continuous categorical Σ-algebra is a cΣa which has an initial object and all colimits of ω-chains, i.e., functors N //C; each functor σC preserves colimits of ω-chains. (N is the linearly ordered set of the nonnegative integers considered as a cat...

This paper shows how systems can be built from their component parts with specified sharing. Its principle contribution is a modular language for configuring systems. A configuration is a description in the new language of how a system is constructed hierarchically from specifications of its component parts. Category theory has been used to represent the composition of specifications that share...

The results presented here first appeared in Chapter 2 of [Mrn1] and were announced with proofs in [MM1]. Our motivation came from [KMS], that introduces the class of direct limits of finite abstract order spaces, a theory due to M. Marshall ([Mar1]). The theory of special groups, introduced in [DM2], is a first-order axiomatization of the algebraic theory of quadratic forms, and there is a nat...

A restriction category is an abstract formulation for a category of partial maps, defined in terms of certain specified idempotents called the restriction idempotents. All categories of partial maps are restriction categories; conversely, a restriction category is a category of partial maps if and only if the restriction idempotents split. Restriction categories facilitate reasoning about parti...

We prove a recognition principle for motivic infinite P1-loop spaces over perfect field. This is achieved by developing theory of framed spaces, which analogue the E-infinity-spaces. A space equipped with transfers along finite syntomic morphisms trivialized cotangent complex in K-theory. Our main result that grouplike are equivalent to full subcategory spectra generated under colimits suspensi...

This paper introduces an expressive class of indexed quotient-inductive types, called QWI within the framework constructive type theory. They are initial algebras for families equational theories with possibly infinitary operators and equations. We prove that types can be derived from quotient inductive in theory toposes natural number object universes, provided those universes satisfy Weakly I...

The group of compactly supported homeomorphisms on a Tychonoff space can be topologized in number ways, including as colimit homeomorphism groups with given compact support, or subgroup the its Stone-\v{C}ech compactification. A is said to have Compactly Supported Homeomorphism Property (CSHP) if these two topologies coincide. authors provide necessary and sufficient conditions for finite produ...