On geometrically transitive Hopf algebroids

Isotopy Quotients of Hopf Algebroids and the Fundamental Groupoid of Digraphs

“…Generators Relations Category of (As numbered in this work) 8), ( 9), (10) left bimodule Surjctive Calculus: (7), ( 8), ( 9), (36 8), ( 9), ( 10), (11), (12) invertible Surjective Calculus: (7), ( 8), ( 9), (36), (38 8), ( 9), ( 10), (11), (12), Π-adaptable (13), (14) bimodule Surjective Calculus: (7), ( 8), ( 9), (36), (38…”

“…The Hopf algebroid structure is defined as follows: (A, C) and H(C) 1 = Hom K−alg (H, C). We refer to Section 3 of [7], for additional details.…”

“…Hence, H iso (C) is precisely the isotopy groupoid of H(C). The notion of isotropy groups was already extended to this setting in Section 5.1 of [7]. 4 The case of surjective calculi…”

Linear Representations and Frobenius Morphisms of Groupoids

“…Let G and H be two groupoids and (X, ϑ, ς) a triple consisting of a set X and two maps ς : X → G 0 , ϑ : X → H 0 . The following definitions are abstract formulations of those given in [16,26] for topological and Lie groupoids, see also [8,10].…”

“…The orbit space of X, is the quotient set X/(H, G) defined using the equivalence relation x ∼ x , if and only if, there exist h ∈ H 1 and g ∈ G 1 with s(h) = ϑ(x) and t(g) = ς(x ), such that hx = x g. Thus it is the set of connected components of the associated two translation groupoid. Next we recall the definition of the tensor product of two groupoid-bisets, see for instance [8,10] or [9]. Fix three groupoids G, H and K. Given (Y, κ, ) and (X, ϑ, ς), a (G, H)-biset and (H, K)biset, respectively.…”