Categorical notions of fibration

Adopting Mental Similarity Notions of Categorical Data Objects to Algorithmic Similarity Functions

The creation of similarity functions based on visual-interactive user feedback is a promising means to capture the Mental Similarity Notion (MSN) in the heads of domain experts. To this end, the user interface (UI) and the algorithmic feedback model (FM) of such interactive systems need to comply with the MSN. The user feedback for categorical attributes shows specific characteristics which we ...

Kant's Four Notions of Freedom

Factorization, Fibration and Torsion

A simple definition of torsion theory is presented, as a factorization system with both classes satisfying the 3–for–2 property. Comparisons with the traditional notion are given, as well as connections with the notions of fibration and of weak factorization system, as used in abstract homotopy theory.

Arithmetic of the [19,1,1,1,1,1] Fibration

Suppose π : S → P is an elliptic K3-surface over C with a section, such that all its singular fibres are multiplicative. Suppose moreover that π is extremal, i.e., the group of sections of π is finite and the rank ρ of the Néron-Severi group NS(S) of S is maximal, so ρ = 20. A formula of Shioda and Tate [Sh90, Cor. 5.3] then implies that π has precisely 6 singular fibers, consisting of ni compo...

Bicategorical Fibration Structures and Stacks

In this paper we introduce two notions —systems of fibrant objects and fibration structures— which will allow us to associate to a bicategory B a homotopy bicategory Ho(B) in such a way that Ho(B) is the universal way to add pseudo-inverses to weak equivalences in B. Furthermore, Ho(B) is locally small when B is and Ho(B) is a 2-category when B is. We thereby resolve two of the problems with kn...