In this paper we introduce a notion of Feynman geometry on which quantum field theories could be properly defined. A strong Feynman geometry is a geometry when the vector space of Aaa structures is finite dimensional. A weak Feynman geometry is a geometry when the vector space of A_∞ structures is infinite dimensional while the relevant operators are of trace-class. We construct families of Feynman geometries with "continuum" as their limit.
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