Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T08:16:56.857Z Has data issue: false hasContentIssue false
  • English
  • Français

When coproducts are biproducts

Published online by Cambridge University Press:  19 February 2016

RICHARD GARNER  and
DANIEL SCHÄPPI
RICHARD GARNER
Affiliation:
Department of Mathematics, Macquarie University, NSW 2109, Australia. e-mail: richard.garner@mq.edu.au
DANIEL SCHÄPPI
Affiliation:
School of Mathematics and Statistics, University of Sheffield, Sheffield, S3 7RH. e-mail: d.schaeppi@sheffield.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Among monoidal categories with finite coproducts preserved by tensoring on the left, we characterise those with finite biproducts as being precisely those in which the initial object and the coproduct of the unit with itself admit right duals. This generalises Houston's result that any compact closed category with finite coproducts admits biproducts.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 1 , July 2016 , pp. 47 - 51
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Barton, R. Semiadditivity and dualizability of 2. Question on MathOverflow, February 2010. http://mathoverflow.net/questions/14402.Google Scholar
[2]Eckmann, B. and Hilton, P. J.Group-like structures in general categories. I. Multiplications and comultiplications. Math. Ann. 145 (1961/1962), 227–255.Google Scholar
[3]Houston, R.Finite products are biproducts in a compact closed category. J. Pure Appl. Alg. 212 (2008), 394–400.CrossRefGoogle Scholar
[4]Mac Lane, S.Categories for the working mathematician. Graduate Texts in Mathematics. vol. 5 (Springer, 1971).Google Scholar