13,420 research outputs found
Skew-closed categories
Spurred by the new examples found by Kornel Szlach\'anyi of a form of lax
monoidal category, the author felt the time ripe to publish a reworking of
Eilenberg-Kelly's original paper on closed categories appropriate to the laxer
context. The new examples are connected with bialgebroids. With Stephen Lack,
we have also used the concept to give an alternative definition of quantum
category and quantum groupoid. Szlach\'anyi has called the lax notion {\em skew
monoidal}. This paper defines {\em skew closed category}, proves Yoneda lemmas
for categories enriched over such, and looks at closed cocompletion.Comment: Version 2 corrects a mistake in axiom (2.4) noticed by Ignacio Lopez
Franco. Only the corrected axiom was used later in the paper so no other
consequential change was needed. A few obvious typos have been corrected.
Some material on weighted colimits, composite modules and skew-promonoidal
categories has been added. Version 3 adds Example 23 and corrects a few
typos.
Serre functors and graded categories
We study Serre structures on categories enriched in pivotal monoidal
categories, and apply this to study Serre structures on two types of graded
k-linear categories: categories with group actions and categories with graded
hom spaces. We check that Serre structures are preserved by taking orbit
categories and skew group categories, and describe the relationship with graded
Frobenius algebras. Using a formal version of Auslander-Reiten translations, we
show that the derived category of a d-representation finite algebra is
fractionally Calabi-Yau if and only if its preprojective algebra has a graded
Nakayama automorphism of finite order. This connects various results in the
literature and gives new examples of fractional Calabi-Yau algebras.Comment: 70 pages; v4 is post referee repor
Configuration spaces of products
We show that the configuration spaces of a product of parallelizable
manifolds may be recovered from those of the factors as the Boardman-Vogt
tensor product of right modules over the operads of little cubes of the
appropriate dimension. We also discuss an analogue of this result for manifolds
that are not necessarily parallelizable, which involves a new operad of skew
little cubes.Comment: 21 pages, 1 figure. To appear in Transactions of the AMS. May vary
slightly from published versio
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