With Fq a finite field of characteristic p, let F(q) be the category whose objects are functors from finite dimensional Fq–vector spaces to F̄p– vector spaces. Extension groups in F(q) can be interpreted as MacLane (or Topological Hochschild) cohomology with twisted coefficients. Furthermore, evaluation on an m dimensional vector space Vm induces a homorphism from Ext F(q) (F, G) to finite group...

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...

Let Q be a finite quiver with vertex set I and arrow set Q1, k a field, and k Q its path algebra with its standard grading. This paper proves some category equivalences involving the quotient category QGr(k Q) := Gr(k Q)/Fdim(k Q) of graded k Q-modules modulo those that are the sum of their finite dimensional submodules, namely QGr(k Q) ≡ ModS(Q) ≡ GrL(Q) ≡ ModL(Q◦)0 ≡ QGr(k Q (n)). Here S(Q) =...

Suppose Hadamard’s criterion applies. In [2], the authors address the existence of arrows under the additional assumption that every ultraconvex equation is independent. We show that B is geometric. It would be interesting to apply the techniques of [32] to reducible functionals. In contrast, the groundbreaking work of A. Wilson on functors was a major advance.

7"o(z) being employed in general to denote the total variation of the function z(t) on /. Thus metrised, (BV) is not a Banach space(2); but it is complete, separable, and boundedly compact. Although a linear space, it is not a "linear topological space" in the sense in which that term is sometimes used, for the topology introduced by the metric (1) is non-uniform. Indeed it is easily seen that ...

In this paper, we introduce the notion of {bf K}-flat projective fuzzy quantales, and give an elementary characterization in terms of a fuzzy binary relation on the fuzzy quantale. Moreover, we  prove that {bf K}-flat projective fuzzy quantales are precisely the coalgebras for a certain comonad on the category of fuzzy quantales. Finally, we present two special cases of {bf K} as examples.

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities Sn−2 ⊂ Sn, n ≥ 5, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial. It is well-known that if the c...

The representations of dimension vector α of the quiver Q can be parametrised by a vector space R(Q,α) on which an algebraic group Gl(α) acts so that the set of orbits is bijective with the set of isomorphism classes of representations of the quiver. We describe the semi–invariant polynomial functions on this vector space in terms of the category of representations. More precisely, we associate...

In this article we introduce the category of noncommutative Artin motives as well as the category of noncommutative mixed Artin motives. In the pure world, we start by proving that the classical category AM(k)Q of Artin motives (over a base field k) can be characterized as the largest category inside Chow motives which fully-embeds into noncommutative Chow motives. Making use of a refined bridg...