Wednesday, September 13, 2023

10:00 a.m.

Zoom Web Conference

PhD Candidate: Hubert Dubé

Supervisor: Kumar Murty

Thesis title: On the Structure of Information Cohomology Exam

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The homological nature of Shannon entropy has been a subject of interest to some for well over fifty years, and yet new approaches are still being suggested studied. Most recently, P. Baudot and D. Bennequin [9], and later J. P. Vigneaux [10] have approached this using topoi whose cohomology of Ext functors contain the entropy function as a 1-cocycle. Moreover, both sets of authors have proven,

under some conditions on their respective constructions, that the entropy in fact generates all of H1. This leaves open questions regarding the general structure (algebraic and categorical) of information structures and the possible higher order cocycles.

This thesis is aimed at extending the theory behind the constructions of J. P. Vigneaux. In particular, we produce new results arising as analogues to results from other cohomology theories, namely the Mayer-Vietoris long exact sequence, to allow for decomposition of information structures into closed information structures, Shapiro’s lemma, to allow for some novel elementary computations, and lastly the Hochschild-Serre spectral sequence, to allow for cohomology computations by means of sub- and quotient structures.

We also provide structural results of algebraic and order-theoretic nature: we provide means to produce useful projective and injective sheaves over information structures, and furthermore prove a general structural result for projective sheaves. This enables easy computations of projective and injective resolutions that provide novel bounds on the cohomological dimension of information structures. Additionally, by taking into account the order-theoretic nature of information structures, we are able to produce an improved bound based on combinatorial invariants.

We finally utilize our results to study the second cohomology groups of some information structures and, in one particular case of interest, find new families of 2-cocycles.

Lastly, we provide with an alternative perspective in which we can view entropy function as a 1-cocycle by investigating it through the lense of operads rather than topoi. We prove that, in this new theory, the Shannon entropy also generates H1.

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The draft of the thesis can be found here: thesis

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