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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 20th 2020
   • PermaLink
   Author: Urs
   Format: MarkdownItexThe term "scalar" used to redirect to _[[ground ring]]_. Hereby I am splitting this off as a little stand-alone entry in order to make room for discussion of the concept of scalars in physics (and then to lead over to things line pseudo-scalars, etc.) <a href="https://ncatlab.org/nlab/revision/scalar/1">v1</a>, <a href="https://ncatlab.org/nlab/show/scalar">current</a>

   The term “scalar” used to redirect to ground ring. Hereby I am splitting this off as a little stand-alone entry in order to make room for discussion of the concept of scalars in physics (and then to lead over to things line pseudo-scalars, etc.)

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeSep 20th 2023
   • (edited Sep 20th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded these pointers to discussion of scalars in the context of ([[dagger-compact closed categories|dagger-]], [[compact closed category|compact-]], [[closed monoidal category|closed]]) [[monoidal categories]], as forming the [[endomorphism ring]] of the [[tensor unit]]: * [[Samson Abramsky]], [[Bob Coecke]], §6 in: *A categorical semantics of quantum protocols*, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04). IEEE Computer Science Press (2004) $[$[arXiv:quant-ph/0402130](http://arxiv.org/abs/quant-ph/0402130), [doi:10.1109/LICS.2004.1319636](https://doi.org/10.1109/LICS.2004.1319636)$]$ * [[Jamie Vicary]], §3 in: *Completeness of dagger-categories and the complex numbers*, J. Math. Phys. **52** (2011) 082104 &lbrack;[arXiv:0807.2927](https://arxiv.org/abs/0807.2927), [doi:10.1063/1.3549117](https://doi.org/10.1063/1.3549117)&rbrack; * [[Jamie Vicary]], §4 in: *Categorical Properties of The Complex Numbers*, Electronic Notes in Theoretical Computer Science **270** 2 (2011) 163-189 &lbrack;[doi:10.1016/j.entcs.2011.01.030](https://doi.org/10.1016/j.entcs.2011.01.030)&rbrack; * [[Chris Heunen]], [[Jamie Vicary]], §2.1 in: *Categories for Quantum Theory*, Oxford University Press 2019 &lbrack;[ISBN:9780198739616](https://global.oup.com/academic/product/categories-for-quantum-theory-9780198739616)&rbrack; based on: [[Chris Heunen]], [[Jamie Vicary]], §2.1 in: *Lectures on categorical quantum mechanics* (2012) &lbrack;[pdf](https://www.cs.ox.ac.uk/files/4551/cqm-notes.pdf), [[HeunenVicary-QuantumLectures.pdf:file]]&rbrack; * [[Chris Heunen]], [[Andre Kornell]], *Axioms for the category of Hilbert spaces*, PNAS **119** 9 (2022) e2117024119 &lbrack;[arXiv:2109.07418](https://arxiv.org/abs/2109.07418), [doi:10.1073/pnas.2117024119](https://doi.org/10.1073/pnas.2117024119)&rbrack; * [[Chris Heunen]], [[Andre Kornell]], [[Nesta van der Schaaf]], *Axioms for the category of Hilbert spaces and linear contractions* &lbrack;[arXiv:2211.02688](https://arxiv.org/abs/2211.02688)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/scalar/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/scalar/3">v3</a>, <a href="https://ncatlab.org/nlab/show/scalar">current</a>

   added these pointers to discussion of scalars in the context of (dagger-, compact-, closed) monoidal categories, as forming the endomorphism ring of the tensor unit:

   • Samson Abramsky, Bob Coecke, §6 in: A categorical semantics of quantum protocols, Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer Science Press (2004) $[$arXiv:quant-ph/0402130, doi:10.1109/LICS.2004.1319636$]$

   • Jamie Vicary, §3 in: Completeness of dagger-categories and the complex numbers, J. Math. Phys. 52 (2011) 082104 [arXiv:0807.2927, doi:10.1063/1.3549117]

   • Jamie Vicary, §4 in: Categorical Properties of The Complex Numbers, Electronic Notes in Theoretical Computer Science 270 2 (2011) 163-189 [doi:10.1016/j.entcs.2011.01.030]

   • Chris Heunen, Jamie Vicary, §2.1 in: Categories for Quantum Theory, Oxford University Press 2019 [ISBN:9780198739616]

     based on:

     Chris Heunen, Jamie Vicary, §2.1 in: Lectures on categorical quantum mechanics (2012) [pdf, HeunenVicary-QuantumLectures.pdf:file]

   • Chris Heunen, Andre Kornell, Axioms for the category of Hilbert spaces, PNAS 119 9 (2022) e2117024119 [arXiv:2109.07418, doi:10.1073/pnas.2117024119]

   • Chris Heunen, Andre Kornell, Nesta van der Schaaf, Axioms for the category of Hilbert spaces and linear contractions [arXiv:2211.02688]

   diff, v3, current