Categories and functors in the representation theory of vertex operator algebras

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In this paper, we use the language of categories to describe representation theory of vertex operator algebras. The category of all vertex operator algebras and the category of modules of a vertex operator algebra are discussed. A homomorphism between two vertex operator algebras should preserve the Virasoro vectors, which is equivalent to commuting with the operators $L(n)$ for all $n\in \mathbb{Z}$. We expand the morphisms of this category so that morphisms are semi-conformal in the sense that they commute with those $L(n)$ with $ n\geq 0$. This expansion does not change the classification of problem and makes the category into a tensor category. The coset construction becomes more natural in this category and relations between the module categories of vertex operator algebras can be described in terms of Hom-functors. As an application, we also construct the corresponding Jacquet functors. The semi-conformal vertex operator subalgebras plays the role of the Levi subgroups of a reductive group.