Recommend "a category of causal-nets" and call for comments and suggestions

Recommend "a category of causal-nets" and call for comments and suggestions

[xuexing lu]

[-- Attachment #1: Type: text/plain, Size: 5012 bytes --]

Hello category theorists,

I am a new man in the field of category theory and I just finished a paper, titled "a category of causal-nets". The paper can be found at (PDF) A category of causal-nets<https://url.au.m.mimecastprotect.com/s/0wsnC1WLjwsMqXYlMULfEHVuIX1?domain=researchgate.net>, or at (1) A category of causal-nets | xuexing lu - Academia.edu<https://url.au.m.mimecastprotect.com/s/zwMRC2xMRkUp8QwgpU1hAH5EDru?domain=academia.edu>. An earlier version can be found at [2201.08963] Causal-net category<https://url.au.m.mimecastprotect.com/s/_h4tC3QNl1Sp2vQOpU2irHQfyb2?domain=arxiv.org> which contains more results.

My purpose in writing this letter is to recommend my work to everyone and to call for comments and suggestions. In this paper, I study the Kleisli category of the "free category on a causal-net" monad. The objects are finite causal-nets (acyclic directed graphs) and the morphisms are functors between free categories generated by causal-nets.

The main result of this paper is a fundamental theorem, which shows an one-to-one correspondence between the six types of fundamental (indecomposable) morphisms and the six basic conventions of graphical calculi. This main result is originally designed as a lemma for proving the existence of causal-net condensation (Theorem 8.0.11, in page 37). Gradually, I find that it is of independent interest and due to its length, it is necessary to write an independent paper.  I recommend this work and hope experts in different fields can find connections with their work. Your comments and suggestions will be helpful for me to write my next paper.

The paper is not very long, but the road to this work is very long. The problem here is that    there are many interesting problems, but solving  them is far beyond my ability.  Even more importantly, this work has revealed many connections among many different fields. So I hope experts from different fields can provide help and guidance. Especially, I hope more people pay attention to these new connections and do more solid work.

The following is my thoughts  about causal-net condensation, some of which may be too physical for mathematicians.

(1) It is a purely categorical framework, totally parallel to that of the factorized homology. The coefficients of factorization homology are $E_n$ algebras, while the coefficients of causal-net condensation are permutation categories. According to the work of Segal and May, the nerve of a permutation category is an $E_\infty$ space whose group completeness is an infinite loop space. From this perspective, causal-net condensation can be seen as a generalized homology theory.
(2) The core basis of causal-net condensation is the discovery that “tensor categories = algebras of coarse-grained monad”. This discovery is not only a natural conclusion of graph calculi theory, but also closely related to topological order theory, tensor network renormalization, and loop quantum gravity.
(3) In the early exploration of causal-net condensation, I was inspired by some ideas in string theory and that of functor of points(Yoneda lemma). I think it's an algebraic version of string theory.
(4) Causal-net condensation can also be regarded as the natural combination of the ideas of string-net condensation and the theory of causal set, which has clearly the property of background independence and provides a new idea for solving the problem of time.
(5) Causal-net condensation is an upgraded version of Baez construction. It reinterprets Baez's graph embedding technique as Kan expansion and introduces the construction of coarse-grainings, which provides a background-independent and non-perturbative framework for quantum field theory.
(6) I find that causal-net condensation has many similarities with some theories in the field of artificial intelligence, such as factor space theory, attribute theory, and granular computing theory. The idea of causal-net condensation is very similar to the ideas of partition logic and quantum logic. I think that it can be understood as a kind of quantum predicate logic, in which the ideas of path integral and resolution principle in computer science, spin networks and Skolem functions are very similar. However, these new findings have not been elaborated in this paper.

Best wishes!
Yours sincerely
Xuexing Lu

Department of math
Zaozhuang University



You're receiving this message because you're a member of the Categories mailing list group from Macquarie University. To take part in this conversation, reply all to this message.

View group files<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=files&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Leave group<https://outlook.office365.com/owa/categories@mq.edu.au/groupsubscription.ashx?source=EscalatedMessage&action=leave&GuestId=4eb9b40c-9b3a-48a5-9781-836e5a171e8b>   |   Learn more about Microsoft 365 Groups<https://aka.ms/o365g>


[-- Attachment #2: Type: text/html, Size: 9991 bytes --]