Recently, symmetric categorical groups are used for the study of the Brauer groups of symmetric monoidal categories. As a part of these efforts, some algebraic structures of the 2-category of symmetric categorical groups SCG are being investigated. In this paper, we consider a 2-categorical analogue of an abelian category, in such a way that it contains SCG as an example. As a main theorem, we ...

This is one of several papers in which we explore the interrelationships among certain categories that can be taken as models for spectra and (highly structured) ring spectra. In this paper, we study certain categories of “diagram ring spectra” that are isomorphic to corresponding categories of “functors with smash product (FSP’s)”. The notion of an FSP was introduced by Bökstedt [2], and his u...

Abstract A well-known question by Gromov asks whether the vanishing of simplicial volume oriented closed aspherical manifolds implies Euler characteristic. We study various versions Gromov’s and collect strategies towards affirmative answers negative to this problem. Moreover, we put into context with other open problems in low- high-dimensional topology. special emphasis is on a comparative an...

We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced. Mathematics Subject Classification (2000). Primary 18D50; Secondary 18G55, 55U35.

We prove a coherence theorem for invertible objects in a symmetric monoidal category. This is used to deduce associativity, skewcommutativity, and related results for multi-graded morphism rings, generalizing the well-known versions for stable homotopy groups.

In a word, sometimes. And it gets harder if the structure on L is not commutative. In this paper we consider the question of what properties are needed on the lattice L equipped with an operation * for several different kinds of categories built using Sets and L to have monoidal and monoidal closed structures. This works best for the Goguen category Set(L) in which membership, but not equality,...

We illustrate a minor error in the biadjointness result for 2-categories of traced monoidal categories and tortile monoidal categories stated by Joyal, Street and Verity. We also show that the biadjointness holds after suitably changing the definition of 2-cells. In the seminal paper “Traced Monoidal Categories” by Joyal, Street and Verity [4], it is claimed that the Int-construction gives a le...

1.1. Let G be an almost simple complex algebraic group, and let GrG be its affine Grassmannian. Recall that if we set O = C[[t]], F = C((t)), then GrG = G(F)/G(O). It is well-known that the subgroup ΩK of polynomial loops into a maximal compact subgroup K ⊂ G projects isomorphically to GrG; thus GrG acquires the structure of a topological group. An algebro-geometric counterpart of this structur...

We define a faithful linear monoidal functor from the partition category, and hence Deligne’s category $\mathrm{\underline{Re}p}(S\_t)$, to additive Karoubi envelope of Heisenberg category. show that induced map on Grothendieck rings is injective corresponds Kronecker coproduct symmetric functions.