Add a reference for cartesian objects and their morphisms.

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added this pointer:

- John Watrous,
*The Theory of Quantum Information*, Cambridge University Press (2018) [doi:10.1017/9781316848142, webpage, pdf]

added pointer to:

- Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki:
*Quantum entanglement*, Rev. Mod. Phys.**81**(2009) 865 [arXiv:quant-ph/0702225, doi:10.1103/RevModPhys.81.865]

added pointer to:

- Ingemar Bengtsson, Karol Życzkowski, Chapter 6 of:
*Geometry of Quantum States — An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [doi:10.1017/CBO9780511535048]

added pointer to:

- Ingemar Bengtsson, Karol Życzkowski, Chapter 15 of:
*Geometry of Quantum States — An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [doi:10.1017/CBO9780511535048]

brief `category:people`

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added pointer to:

- Ingemar Bengtsson, Karol Życzkowski,
*Geometry of Quantum States — An Introduction to Quantum Entanglement*, Cambridge University Press (2006) [doi:10.1017/CBO9780511535048]

and will add this to a few related entries, too (such as to quantum state, quantum entanglement, …)

]]>This partly answers the first question in #24:

They don’t have an actual type theory yet.

It’s the same situation as with my suggestion of linear homotopy type theory in the past: A semantics neatly organized by simple rules which seem to lend themselves to type-theoretic formalization, but no actual formal syntax yet.

]]>expanded this out to the following dialogue, highlighting that actual type-theoretic syntax (inference rules) for this intended semantics remains to be given:

[Cisinski in video 3 at 1:27:43]: I won’t provide the full syntax yet and actually I would be very happy to discuss that, because we don’t know yet and I have questions myself, actually.

[Awodey in video 3 at 1:46:23]: Maybe I’ll suggest something, you tell me if you agree: What we have is a kind of axiomatization of the semantics of a system for type theory, so that we know what exactly we want formalize in the type theory, and what depends on what, and it articulates and structures the intended interpretation of the type theory in a very useful way. Maybe in the way that the axiomatic description of a cartesian closed category was very good to have for formulating the lambda-calculus. But I think that what we have is more on the side of the axiomatic description of the semantics, like the cartesian closed category, that it is on the side of the lambda-calculus itself. So, maybe I would suggest the term “abstract type theory” to describe this system as an intermediate in between an actual formally implemented system of type theory and the big unclear world of possible semantics and all the different structures that one could try to capture with a type theory, in between is this abstract type theory which specifies a particular structure that we want to capture in our type theory, which is a very very useful methodological step. […] I am trying to maybe reconcile:

Some people would prefer to call a type theory only something which can immediately be implemented in a computer. So that’s different than an abstract description of a structure that we would want to describe in such a type theory.

[Cisinski in video 3 at 1:49:28]: I agree with what you say but I still have the hope to be able to produce an actual syntax […] that’s really the goal.

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added this quote:

]]>[video 3 at 1:27:43]: I won’t provide the full syntax yet and actually I would be very happy to discuss that, because we don’t know yet and I have questions myself, actually.

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added this remark:

This is based on the discussion of straightening and unstraightening entirely within the context of quasi-categories from

- Denis-Charles Cisinski, Hoang Kim Nguyen,
*The universal coCartesian fibration*[arXiv:2210.08945]

which (along the lines of the discussion of the universal left fibration from Cisinski 2019) allows to understand the universal coCartesian fibration as categorical semantics for the univalent type universe in directed homotopy type theory

(see video 3 at 1:16:58 and slide 3.33).

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added these pointers:

Discussion of straightening and unstraightening entirely within the context of quasi-categories:

- Denis-Charles Cisinski, Hoang Kim Nguyen,
*The universal coCartesian fibration*[arXiv:2210.08945]

which (along the lines of the discussion of the universal left fibration from Cisinski 2019) allows to understand the universal coCartesian fibration as categorical semantics for the univalent type universe in directed homotopy type theory:

- {#CisinskiEtAl23} Denis-Charles Cisinski, Hoang Kim Nguyen, Tashi Walde:
*Univalent Directed Type Theory*, lecture series in the*CMU Homotopy Type Theory Seminar*(13, 20, 27 Mar 2023) [web, video 1:YT, 2:YT, 3:YT; slides 0:pdf, 1:pdf, 2:pdf, 3:pdf]

(see video 3 at 1:16:58 and slides 3.33).

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added pointer to:

- Simon Burton,
*A Short Guide to Anyons and Modular Functors*$[$arXiv:1610.05384$]$

made some trivial additions

]]>[just as a technical aside: to get your code for hyperlinks rendered here on the nForum, be sure that the button “Markdown+Itex” below the edit pane is activated ]

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