My Introduction to Homotopy Type Theory textbook is finished and on the ArXiv
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Egbert Rijke
Dec 23, 2022, 4:54:39 AM12/23/22
to Homotopy Type Theory
Dear homotopy type theorists,
My textbook Introduction to Homotopy Type Theory is finished and available on the ArXiv:
https://arxiv.org/abs/2212.11082
From the abstract:
This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice to consider equivalent objects to be the same, for example, to identify isomorphic groups. In set theory it is not possible to make this common practice formal. For example, there are as many distinct trivial groups in set theory as there are distinct singleton sets. Type theory, on the other hand, takes a more structural approach to the foundations of mathematics that accommodates the univalence axiom. This, however, requires us to rethink what it means for two objects to be equal. This textbook introduces the reader to Martin-Löf's dependent type theory, to the central concepts of univalent mathematics, and shows the reader how to do mathematics from a univalent point of view. Over 200 exercises are included to train the reader in type theoretic reasoning. The book is entirely self-contained, and in particular no prior familiarity with type theory or homotopy theory is assumed.
My textbook Introduction to Homotopy Type Theory is finished and available on the ArXiv:
https://arxiv.org/abs/2212.11082
From the abstract:
This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice to consider equivalent objects to be the same, for example, to identify isomorphic groups. In set theory it is not possible to make this common practice formal. For example, there are as many distinct trivial groups in set theory as there are distinct singleton sets. Type theory, on the other hand, takes a more structural approach to the foundations of mathematics that accommodates the univalence axiom. This, however, requires us to rethink what it means for two objects to be equal. This textbook introduces the reader to Martin-Löf's dependent type theory, to the central concepts of univalent mathematics, and shows the reader how to do mathematics from a univalent point of view. Over 200 exercises are included to train the reader in type theoretic reasoning. The book is entirely self-contained, and in particular no prior familiarity with type theory or homotopy theory is assumed.
Over Christmas I will write a blog post in which I will go more into the content of the book. For now: Enjoy!
Happy holidays to everyone!
Egbert
João Alves Silva Júnior
Dec 23, 2022, 9:22:56 AM12/23/22
to Egbert Rijke, Homotopy Type Theory
Thank you!
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EMILY RIEHL
Dec 23, 2022, 5:52:02 PM12/23/22
to HomotopyT...@googlegroups.com
Congratulations Egbert!
Last fall I taught a topics course on homotopy type theory following this book to an audience of mostly mathematics students. The weekly problem sets were all in Agda, drawing inspiration from Egbert's excellent lists of exercises and extensive formalized library. Course materials are available here: https://github.com/emilyriehl/721
I mention all of this to say that I had an excellent experience teaching using Egbert's book. I had to skip several topics due to time constraints but I never found myself wishing the material were presented in a different order, which is quite remarkable. Kudos Egbert!
Emily
Professor of Mathematics (she/her)
Johns Hopkins University
emilyriehl.github.io
On 12/23/22 04:54, Egbert Rijke wrote:
> Dear homotopy type theorists,
>
> My textbook Introduction to Homotopy Type Theory is finished and available on the ArXiv:
>
> https://arxiv.org/abs/2212.11082 <https://nam02.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fabs%2F2212.11082&data=05%7C01%7Ceriehl%40jhu.edu%7C0aa33ca5d7df4698161908dae4cbb6a2%7C9fa4f438b1e6473b803f86f8aedf0dec%7C0%7C0%7C638073860853350577%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=aVyJRuhCTZwnztt1H3l2aQMOvn2sfNUVD%2B0obnR%2FC0Y%3D&reserved=0>
> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGqv1ODuH3xtsFyFEekjwNH4SUN%2BdU8B1te2ZLX%2BNZsQLeChxg%40mail.gmail.com <https://nam02.safelinks.protection.outlook.com/?url=https%3A%2F%2Fgroups.google.com%2Fd%2Fmsgid%2FHomotopyTypeTheory%2FCAGqv1ODuH3xtsFyFEekjwNH4SUN%252BdU8B1te2ZLX%252BNZsQLeChxg%2540mail.gmail.com%3Futm_medium%3Demail%26utm_source%3Dfooter&data=05%7C01%7Ceriehl%40jhu.edu%7C0aa33ca5d7df4698161908dae4cbb6a2%7C9fa4f438b1e6473b803f86f8aedf0dec%7C0%7C0%7C638073860853350577%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C3000%7C%7C%7C&sdata=yP5HBchPgpQNsZlwcNVENWmOKZeHdVBHkSneMgvzjCc%3D&reserved=0>.
Last fall I taught a topics course on homotopy type theory following this book to an audience of mostly mathematics students. The weekly problem sets were all in Agda, drawing inspiration from Egbert's excellent lists of exercises and extensive formalized library. Course materials are available here: https://github.com/emilyriehl/721
I mention all of this to say that I had an excellent experience teaching using Egbert's book. I had to skip several topics due to time constraints but I never found myself wishing the material were presented in a different order, which is quite remarkable. Kudos Egbert!
Emily
Professor of Mathematics (she/her)
Johns Hopkins University
emilyriehl.github.io
On 12/23/22 04:54, Egbert Rijke wrote:
> Dear homotopy type theorists,
>
> My textbook Introduction to Homotopy Type Theory is finished and available on the ArXiv:
>
>
> From the abstract:
> This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice to consider equivalent objects to be the same, for example, to identify isomorphic groups. In set theory it is not possible to make this common practice formal. For example, there are as many distinct trivial groups in set theory as there are distinct singleton sets. Type theory, on the other hand, takes a more structural approach to the foundations of mathematics that accommodates the univalence axiom. This, however, requires us to rethink what it means for two objects to be equal. This textbook introduces the reader to Martin-Löf's dependent type theory, to the central concepts of univalent mathematics, and shows the reader how to do mathematics from a univalent point of view. Over 200 exercises are included to train the reader in type
> theoretic reasoning. The book is entirely self-contained, and in particular no prior familiarity with type theory or homotopy theory is assumed.
>
> Over Christmas I will write a blog post in which I will go more into the content of the book. For now: Enjoy!
>
> Happy holidays to everyone!
> Egbert
>
> From the abstract:
> This is an introductory textbook to univalent mathematics and homotopy type theory, a mathematical foundation that takes advantage of the structural nature of mathematical definitions and constructions. It is common in mathematical practice to consider equivalent objects to be the same, for example, to identify isomorphic groups. In set theory it is not possible to make this common practice formal. For example, there are as many distinct trivial groups in set theory as there are distinct singleton sets. Type theory, on the other hand, takes a more structural approach to the foundations of mathematics that accommodates the univalence axiom. This, however, requires us to rethink what it means for two objects to be equal. This textbook introduces the reader to Martin-Löf's dependent type theory, to the central concepts of univalent mathematics, and shows the reader how to do mathematics from a univalent point of view. Over 200 exercises are included to train the reader in type
> theoretic reasoning. The book is entirely self-contained, and in particular no prior familiarity with type theory or homotopy theory is assumed.
>
> Over Christmas I will write a blog post in which I will go more into the content of the book. For now: Enjoy!
>
> Happy holidays to everyone!
> Egbert
>
> --
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Urs Schreiber
Jan 3, 2023, 2:16:40 PM1/3/23
to Egbert Rijke, Homotopy Type Theory
Hi Egbert,
nice to see your notes now available in a stably referenceable way!
They could fill quite a few gaps that the existing textbook literature
leaves open.
On that note, it seems that a fair bit of material has been removed in
the arXiv version?
(Maybe to make room for large margins?)
For referencing on the nLab I now find myself pointing mainly to the
version of your notes from 2018 (these here:
https://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), which
have discussion for instance of homotopy pullbacks/pushouts that seem
to have later been dropped, together with much material depending on
these notions (if I am seeing this correctly ?)
I can imagine this is at least in large part the publisher's decision,
but just to say that if there is any wiggle room left, then I would
think it most worthwhile if these topics could make it into the final
book version.
All my best wishes for the New Year,
Urs
> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGqv1ODuH3xtsFyFEekjwNH4SUN%2BdU8B1te2ZLX%2BNZsQLeChxg%40mail.gmail.com.
nice to see your notes now available in a stably referenceable way!
They could fill quite a few gaps that the existing textbook literature
leaves open.
On that note, it seems that a fair bit of material has been removed in
the arXiv version?
(Maybe to make room for large margins?)
For referencing on the nLab I now find myself pointing mainly to the
version of your notes from 2018 (these here:
https://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), which
have discussion for instance of homotopy pullbacks/pushouts that seem
to have later been dropped, together with much material depending on
these notions (if I am seeing this correctly ?)
I can imagine this is at least in large part the publisher's decision,
but just to say that if there is any wiggle room left, then I would
think it most worthwhile if these topics could make it into the final
book version.
All my best wishes for the New Year,
Urs
> --
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
> To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeThe...@googlegroups.com.
> You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group.
> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAGqv1ODuH3xtsFyFEekjwNH4SUN%2BdU8B1te2ZLX%2BNZsQLeChxg%40mail.gmail.com.
Jon Sterling
Jan 3, 2023, 5:06:11 PM1/3/23
to Urs Schreiber, Egbert Rijke, Homotopy Type Theory
Hi all,
I second Urs's congratulations on this great achievement! Just want to also add that I likewise miss the material on descent & flattening, for which your old notes provided an excellent reference. Of course, perhaps this leaves room for an "Advanced Topics in Homotopy Type Theory" followup ;-)
Best,
Jon
> To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CA%2BKbugeyU16TCLH9MbiBPUQHFiqytnpftQ%3DFmPr8zLSzk1ODHA%40mail.gmail.com.
I second Urs's congratulations on this great achievement! Just want to also add that I likewise miss the material on descent & flattening, for which your old notes provided an excellent reference. Of course, perhaps this leaves room for an "Advanced Topics in Homotopy Type Theory" followup ;-)
Best,
Jon
Madeleine Birchfield
Jan 13, 2023, 9:55:44 AM1/13/23
to Homotopy Type Theory
Congratulations on finishing this textbook, Egbert! This textbook, in its draft format, has been very useful in my learning of homotopy type theory as a foundations of mathematics back a few years ago, and I'm glad to see it finished and hopefully published and available in hardcover format soon.
As for the splitting of the textbook in two, I actually agree with the decision to remove the material on using homotopy type theory for synthetic homotopy theory from the textbook, so that this textbook focuses on the foundational aspects of homotopy type theory. However, I do have to agree with Jon that I hope the second half of the textbook gets published as its own textbook; my preferred title would probably be "Synthetic Homotopy Theory" since much of the removed material was specifically about synthetic homotopy theory.
Madeleine Birchfield
As for the splitting of the textbook in two, I actually agree with the decision to remove the material on using homotopy type theory for synthetic homotopy theory from the textbook, so that this textbook focuses on the foundational aspects of homotopy type theory. However, I do have to agree with Jon that I hope the second half of the textbook gets published as its own textbook; my preferred title would probably be "Synthetic Homotopy Theory" since much of the removed material was specifically about synthetic homotopy theory.
Thanks,
Madeleine Birchfield
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