2022/08/30 by Fiona Torzewska, Torzewska, Fiona
Mathematics · Medicine · #18M15 #57N35 #57R56 #Advanced Topics in Algebra #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Ophthalmology and Eye Disorders #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2208.14504
openalex publication_date 2022/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We construct a category HomCob whose objects are \it homotopically 1-finitely generated topological spaces, and whose morphisms are \it cofibrant cospans. Given a manifold submanifold pair (M,A), we prove that there exists functors into HomCob from the full subgroupoid of the mapping class groupoid MCGMA, and from the full subgroupoid of the motion groupoid MotMA, whose objects are homotopically 1-finitely generated. We also construct a family of functors ZG\colon HomCob→ Vect, one for each finite group G. These generalise topological quantum field theories previously constructed by Yetter, and an untwisted version of Dijkgraaf-Witten. Given a space X, we prove that ZG(X) can be expressed as the ℂ-vector space with basis natural transformation classes of maps \π(X,X0)→ G\ for some finite representative set of points X0⊂ X, demonstrating that ZG is explicitly calculable.