A general definition of dependent type theories

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The preprint version of the paper A overall definition of dependent kind
theories
has in the slay appeared on the arXiv! Over three
years previously Peter Lumsdaine invited me to work on the
topic, which I gladly accredited, and dragged my scholar Philipp
Haselwarter
into it. We discipline out to present an answer to
the queation:

What is kind concept, precisely?

Now not not up to for me the motivation to work on such a thankless topic came from Vladimir
Voevodsky, who would quiz the request to kind-theoretic audiences. Some took him to be a
troll and others a newcomer who correct needed to learn extra kind concept. I become amongst the
latter, but in the slay the request bought thru to me – I could well well furthermore new any form of
explicit examples of kind theories, but not a comprehensive and mathematically real
definition of the overall concept.

It’s miles simply too straightforward to omit the request by claiming that kind concept is an begin-ended concept
which subsequently can not be completely captured by any mathematical definition. Of
route it is begin-ended, but it completely would not apply the least bit that we ought to not even
attempt to elucidate it. If geometers had been equally fatalistic relating to the begin-ended belief of
rental we would by no manner grasp had popular geometry, topology, sheaves – heck, half of 20th
century arithmetic would not be there!

Useless to claim, we are neither the first nor the last to present a definition of kind concept, and
that’s how things want to be. We allege no precedence or supremacy over other definitions and
views of kind concept. Our come could well well furthermore possibly be described as “concrete” and
“proof-theoretic”:

  1. We desired to protect end to mature syntax.
  2. We gave a whole and real definition.
  3. We aimed for a level of generality that enables functional meta-concept of a huge series of kind theories.

One can argue every of the above ingredients, and we now grasp got completed so amongst ourselves time and again.
Nonetheless, I feel that we now grasp got completed something priceless – however the closing
judges will probably be our readers, or lack of them. You have to well well furthermore be suitable invited to verify out the
paper.

Download PDF: arxiv.org/pdf/2009.05539.pdf

I ought to not omit to point out that Peter, with modest relieve from Philipp and me,
formalized nearly your whole paper in Coq! Witness the repository
overall-kind-theories at
GitHub.

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