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This article is a follow-up of ``Holonomy and Path Structures in GeneralRelativity and Yang-Mills Theory" by Barrett, J. W. (Int.J.Theor.Phys., vol.30,No.9, 1991). Its main goal is to provide an alternative proof of this part ofthe reconstruction theorem which concerns the existence of a connection. Aconstruction of connection 1-form is presented. The formula expressing thelocal coefficients of connection in terms of the holonomy map is obtained as animmediate consequence of that construction. Thus the derived formula coincideswith that used in "On Loop Space Formulation of Gauge Theories" by Chan, H.-M.,Scharbach, P. and Tsou S.T. (Ann.Phys., vol.167, 454-472, 1986). Thereconstruction and representation theorems form a generalization of the factthat the pointed configuration space of the classical Yang-Mills theory isequivalent to the set of all holonomy maps. The point of this generalization isthat there is a one-to-one correspondence not only between the holonomy mapsand the orbits in the space of connections, but also between all maps from theloop space on $M$ to group $G$ fulfilling some axioms and all possibleequivalence classes of $P(M,G)$ bundles with connection, where the equivalencerelation is defined by bundle isomorphism in a natural way.Comment: amslatex, 7 pages, no figure

Axiomatic holonomy maps and generalized Yang-Mills moduli space