A bstract In this work, we show that a critical point of 1d self-dual boundary phase transition between two gapped boundaries the ℤ N topological order can be described by mathematical structure called an enriched fusion category. The viewed as gappable non-chiral gapless order. theory 2d orders developed Kong and Zheng (arXiv:1905.04924 arXiv:1912.01760) tells us all macroscopic observables on...

Directed Algebraic Topology is a recent field, deeply linked with Category Theory. A ‘directed space’ has directed homotopies (generally non reversible), directed homology groups (enriched with a preorder) and fundamental n-categories (replacing the fundamental ngroupoids of the classical case). On the other hand, directed homotopy can give geometric models for lax higher categories. Applicatio...

Each distributor between categories enriched over a small quantaloid Q gives rise to two adjunctions between the categories of contravariant and covariant presheaves, and hence to two monads. These two adjunctions are respectively generalizations of Isbell adjunctions and Kan extensions in category theory. It is proved that these two processes are functorial with infomorphisms playing as morphi...

For a small category K enriched over a suitable monoidal category V , the free completion of K under colimits is the presheaf category [K ,V ]. If K is large, its free completion under colimits is the V -category PK of small presheaves on K , where a presheaf is small if it is a left Kan extension of some presheaf with small domain. We study the existence of limits and of monoidal closed struct...

This article represents a preliminary attempt to link Kan extensions, and some of their further developments, to Fourier theory and quantum algebra through ∗autonomous monoidal categories and related structures. There is a close resemblance to convolution products and the Wiener algebra (of transforms) in functional analysis. The analysis term “kernel” (of a distribution) has also been adapted ...

Abstract We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind enriched category. The base for enrichment is category commutative monoids—or in straightforward generalisation, modules over rig k . However, tensor product on this not usual one, but rather warping it by certain monoidal comonad Q Thus sense, skew sense Szlachányi. Our first main result ...

background the relationship between biomarkers and imaging features is important because imaging findings can predict molecular features. objectives to investigate the relationship between clinicopathologic and radiologic factors and the immunohistochemical (ihc) profiles associated with breast cancer. patients and methods from december 2004 to september 2013, 200 patients (mean age, 56 years; ...

‎This paper develops a basic theory of $H$-groups‎. ‎We‎ ‎introduce a special quotient of $H$-groups and‎ ‎extend some algebraic constructions of topological groups to the category‎ ‎of H-groups and H-maps and then present a functor from this category to the category of quasitopological groups‎.