String diagrams for traced and compact categories are oriented 1-cobordisms
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TracedCobAlg.pdf
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513.62 KB
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Author(s)
Spivak, David I
Schultz, Patrick M
Date Issued
2017
Journal
Journal of Pure and Applied Algebra
Publisher
Elsevier
Citation
Spivak, David I., Patrick Schultz and Dylan Rupel. "String diagrams for traced and compact categories are oriented 1-cobordisms." Journal of Pure and Applied Algebra 221, no. 8 (2017): pp. 2064-2110.
Version
Author's final manuscript
Abstract
String diagrams for traced and compact categories are oriented 1-cobordisms David I. Spivak ∗ Patrick Schultz ∗ Massachusetts Institute of Technology, Cambridge, MA 02139 Dylan Rupel † , ‡ Northeastern University, Boston, MA 02115 Abstract We give an alternate conception of string diagrams as labeled 1-dimensional oriented cobordisms, the operad of which we denote by Cob / O , where O is the set of string labels. The axioms of traced (symmetric monoidal) categories are fully en- coded by Cob / O in the sense that there is an equivalence between Cob / O -algebras, for varying O , and traced categories with varying object set. The same holds for compact (closed) categories, the difference being in terms of variance in O . As a consequence of our main theorem, we give a characterization of the 2-category of traced categories solely in terms of those of monoidal and compact categories, without any reference to the usual structures or axioms of traced categories. In an appendix we offer a complete proof of the well-known relationship between the 2-category of monoidal categories with strong monoidal functors and the 2-category of monoidal categories whose object set is free with strict functors; similarly for traced and compact categories.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Creative Commons Attribution-NonCommercial-NoDerivs License
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DOI of Published Version
https://doi.org/10.1016/j.jpaa.2016.10.009
