In this short note we construct two families of examples large stratifying systems in module categories algebras. The first consist on infinite size the category an algebra A. second family show that a finite system dimensional A can be arbitrarily comparison to number isomorphism classes simple A-modules. We both are built using well-established results higher homological algebra.

In this paper we explore the isotropic stable motivic homotopy category constructed from usual category, following work of Vishik on motives (see [28]), by killing anisotropic varieties. particular, focus cohomology operations in realm and study structure Steenrod algebra. Then, construct an version Adams spectral sequence apply it to find a complete description groups sphere spectrum, which ha...

By the SYZ construction, a mirror pair (X,Xˇ) of complex torus X and partner Xˇ is described as special Lagrangian fibrations X→B Xˇ→B on same base space B. Then, by transform, we can construct simple projectively flat bundle from each affine multisection with unitary local system along it. However, there are ambiguities choices transition functions it, this causes difficulties when try to func...

We consider Frobenius objects in the category Span, where are sets and morphisms isomorphism classes of spans sets. show that such structures correspondence with data can be characterized terms simplicial An interesting class examples comes from groupoids. Our primary motivation is Span viewed as a set-theoretic model for symplectic category, thus provide models classical topological field theo...

Grothendieck duality theory assigns to essentially-finite-type maps f of noetherian schemes a pseudofunctor f× right-adjoint to Rf∗, and a pseudofunctor f ! agreeing with f× when f is proper, but equal to the usual inverse image f∗ when f is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken...

It is unknown whether two graphs can be tested for isomorphism in polynomial time. A classical approach to the Graph Isomorphism Problem is the d-dimensional Weisfeiler-Lehman algorithm. The d-dimensional WL-algorithm can distinguish many pairs of graphs, but the pairs of non-isomorphic graphs constructed by Cai, Fürer and Immerman it cannot distinguish. If d is fixed, then the WL-algorithm run...

We consider definably compact groups in an o-minimal expansion of a real closed field. It is known that to each such group G is associated a natural exact sequence 1→ G00 → G → G/G00 → 1 where G00 is the “infinitesimal subgroup” of G and G/G00 is a compact real Lie group. We show that given a connected open subset U of G/G00 there is a canonical isomorphism between the fundamental group of U an...

A functor G : C → D is said to preserve limits of a diagram D : I → C if it sends any limiting cone from x to D to a limiting cone from G(x) to G◦D. When G preserves limits of a diagram D this entails directly that there is an isomorphism G(lim ←−ID) ∼= lim ←−I(G ◦D) between objects. In general, such an isomorphism alone is not sufficient to ensure that G preserves limits. This paper shows how,...

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commu...