  John C.Baez reinterpreted 2-dimensional and 3-dimensional topological quantum
field theories (abbreviated as 2-TQFT and 3-TQFT) in "A prehistory of
n-categorical physics"[JC11]. Inspired by his idea, this paper utilizes
cochains to prove some properties of 2-TQFT and 3-TQFT. We also prove that
cochains form an $A_{\infty}$ algebra under certain conditions.
