White Rose University Consortium logo
University of Leeds logo University of Sheffield logo York University logo

Day convolution for monoidal bicategories

Corner, Alexander S. (2016) Day convolution for monoidal bicategories. PhD thesis, University of Sheffield.

[img]
Preview
Text
Alexander_Corner_PhD_Thesis.pdf
Available under License Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 UK: England & Wales.

Download (843Kb) | Preview

Abstract

Ends and coends can be described as objects which are universal amongst extranatural transformations. We describe a cate- gorification of this idea, extrapseudonatural transformations, in such a way that bicodescent objects are the objects which are universal amongst such transfor- mations. We recast familiar results about coends in this new setting, providing analogous results for bicodescent objects. In particular we prove a Fubini theorem for bicodescent objects. The free cocompletion of a category C is given by its category of presheaves [C^op ,Set]. If C is also monoidal then its category of presheaves can be pro- vided with a monoidal structure via the convolution product of Day. This monoidal structure describes [C^op ,Set] as the free monoidal cocompletion of C. Day’s more general statement, in the V-enriched setting, is that if C is a promonoidal V-category then [C^op ,V] possesses a monoidal structure via the convolution product. We define promonoidal bicategories and go on to show that if A is a promonoidal bicategory then the bicategory of pseudofunctors Bicat(A^op ,Cat) is a monoidal bicategory.

Item Type: Thesis (PhD)
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Alexander S. Corner
Date Deposited: 31 Mar 2017 13:57
Last Modified: 31 Mar 2017 13:57
URI: http://etheses.whiterose.ac.uk/id/eprint/16767

Actions (repository staff only: login required)