Discrete and Conservative Factorizations in Fib(B)

We focus on the transfer of some known orthogonal factorization systems from $$\mathsf {Cat}$$ Cat to the 2-category $${\mathsf {Fib}}(B)$$ Fib ( B ) of fibrations over a fixed base category B: the internal version of the comprehensive factorization, and the factorization systems given by (sequence of coidentifiers, discrete morphism) and (sequence of coinverters, conservative morphism) respectively. For the class of fibrewise opfibrations in $${\mathsf {Fib}}(B)$$ Fib ( B ) , the construction of the latter two simplify to a single coidentifier (respectively coinverter) followed by an internal discrete opfibration (resp. fibrewise opfibration in groupoids). We show how these results follow from their analogues in $$\mathsf {Cat}$$ Cat , providing suitable conditions on a 2-category $${\mathcal {C}}$$ C , that allow the transfer of the construction of coinverters and coidentifiers from $${\mathcal {C}}$$ C to $${\mathsf {Fib}}_{{\mathcal {C}}}(B)$$ Fib C ( B ) .

Modal descent

Any modality in homotopy type theory gives rise to an orthogonal factorization system of which the left class is stable under pullbacks. We show that there is a second orthogonal factorization system associated with any modality, of which the left class is the class of ○-equivalences and the right class is the class of ○-étale maps. This factorization system is called the modal reflective factorization system of a modality, and we give a precise characterization of the orthogonal factorization systems that arise as the modal reflective factorization system of a modality. In the special case of the n-truncation, the modal reflective factorization system has a simple description: we show that the n-étale maps are the maps that are right orthogonal to the map $${\rm{1}} \to {\rm{ }}{{\rm{S}}^{n + 1}}$$ . We use the ○-étale maps to prove a modal descent theorem: a map with modal fibers into ○X is the same thing as a ○-étale map into a type X. We conclude with an application to real-cohesive homotopy type theory and remark how ○-étale maps relate to the formally etale maps from algebraic geometry.

Indexed type theories

In this paper, we define indexed type theories which are related to indexed (∞-)categories in the same way as (homotopy) type theories are related to (∞-)categories. We define several standard constructions for such theories including finite (co)limits, arbitrary (co)products, exponents, object classifiers, and orthogonal factorization systems. We also prove that these constructions are equivalent to their type theoretic counterparts such as Σ-types, unit types, identity types, finite higher inductive types, Π-types, univalent universes, and higher modalities.

On some orthogonal factorization systems

The orthogonality relation between arrows in the class of all morphisms of a given category C yields a "concrete" antitone Galois connection between the class of all subclasses of morphisms of C. For a class Σ of morphisms of C, we denote by ⊥Σ (resp., Σ⊥) the class of all morphisms f in C such that f ⊥ g (resp., g ⊥ f) for each morphism g in Σ. A couple (Σ, Γ) of classes of morphisms is said to be an (orthogonal) prefactorization system if If, in addition the pfs satisfies then it will be called a dense prefactorization system. A pair [Formula: see text] of classes of morphisms in a category C is called an (orthogonal) factorization system if it is a prefactorization system and each morphism f in C has a factorization f = me, with [Formula: see text] and [Formula: see text]. This paper provides several examples of factorization systems and dense factorization systems in the category Top of topological spaces.

Optimization of the NDV-N-2-HACC/CMC Microspheres Preparation

Purpose:in this study, the conditions of the preparation of NDV-N-2-HACC/CMC microspheres are optimized. Methods:Using entrapment efficiency, particle size, Zeta potential as the evaluating indicators, 3 main factors to influence the preparation for microspheres were optimized by orthogonal factorization method. Result:The optimal conditions were 1.0 mg/ml HACC, 1.2 mg/ml CMC, 1:3 (v/v) NDV/HACC, and 1200 r/min and 30 min for stirring. The range of particle size was 192.1-595.2 nm, and average size was 304.3 nm. Zeta potential was +32.50 mV; encapsulation efficiency (EE) was (98.96±2.1) %.Conclusion:the conditions of the preparation of NDV-N-2-HACC/CMC microspheres are optimized.

Robust Filtering for Linear Equality Constrained Systems

This paper deals with the robust filtering problem for linear discrete-time constrained systems. The purpose is the design of a linear filter such that the resulting error system is bounded. An orthogonal factorization is used to decompose the original robust filtering problem into stochastic and deterministic parts, which are then solved separately. Finally, a numerical example is presented to demonstrate the applicability of the proposed method.