On Two-Dimensional Closed–Open Topological Field Theories

Topological field theories (TFTs) have captured the attention of mathematicians due to their various applications. In categorical terms, an nTFT is defined as a monoidal functor that maps the category of n-dimensional cobordisms to the category of vector spaces. In this paper, we introduce the category of two-dimensional closed–open cobordisms, denoted as 2CobCO. We demonstrate that the generating morphisms in this category total 35. Furthermore, we establish that the category 2CobCO is a monoidal category. We define a triple (B,ε,ε′) as a doubly Frobenius algebra if both (B,ε) and (B,ε′) are Frobenius algebras. We then introduce the category of doubly Frobenius algebras, wherein the objects are doubly Frobenius algebras and the morphisms are homomorphisms of Frobenius algebras that satisfy specific compatibility conditions. Additionally, we present a new type of 2TFT, which we refer to as the two-dimensional closed–open TFT (denoted as 2TFTCO). We demonstrate that the category of all 2TFTCO, referred to as 2TFTCO, is equivalent to the category of all commutative doubly Frobenius algebras, denoted as CF.