2021/04/26 by Léonard Guetta, Guetta, Léonard
Mathematics · Medicine · #18G99 #18N10 #18N30 #18N40 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications
paper · pdf · doi:10.48550/arxiv.2104.12662
openalex publication_date 2021/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this dissertation, we compare the "classical" homology of an\n\ω-category (defined as the homology of its Street nerve) with its\npolygraphic homology. More precisely, we prove that both homologies generally\ndo not coincide and call homologically coherent the particular strict\n\ω-categories for which polygraphic homology and homology of the nerve do\ncoincide. The goal pursued is to find abstract and concrete criteria to detect\nhomologically coherent \ω-categories. For example, we prove that all\n(small) categories, considered as strict \ω-categories with unit cells\nabove dimension 1, are homologically coherent. We also introduce the notion of\nbubble-free 2-category and conjecture that a cofibrant 2-category is\nhomologically coherent if and only if it is bubble-free. We also prove\nimportant results concerning free strict \ω-categories on polygraphs\n(also known as computads), such as the fact that if F is a discrete Conduch 'e\n\ω-functor from C to D and if D is a free strict \ω-category on a\npolygraph, then so is C. Overall, this thesis achieves to build a general\nframework in which to study the homology of strict \ω-categories using\ntools of abstract homotopical algebra such as Quillen's theory of model\ncategories or Grothendieck's theory of derivators.\n