2025/05/28 by Hoffbeck, Eric, Leray, Johan, Vallette, Bruno · 2 citations
#14D15 #16T10 #17B55 #18M70 #18M85 #18N40 #18N50 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Quantum Algebra (math.QA)
paper · doi:10.48550/arxiv.2505.22004
In this paper, we settle the homotopy properties of the infinity-morphisms of homotopy (bial)-gebras over properads, i.e. algebraic structures made up of operations with several inputs and outputs. We start by providing the literature with characterizations for the various types of infinity-morphisms, the most seminal one being the equivalence between infinity-quasi-isomorphisms and zig-zags of quasi-isomorphisms which plays a key role in the study the formality property. We establish a simplicial enrichment for the categories of gebras over some cofibrant properads together with their infinity-morphisms, whose homotopy category provides us with the localisation with respect to infinity-quasi-isomorphisms. These results extend to the properadic level known properties for operads, but the lack of the rectification procedure in this setting forces us to use different methods.