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The colored parenthesized braid operad is a sequence of full subcategories ofthe fundamental groupoids of the configuration spaces of the complex plane, anda conformal block, a physical quantity of chiral two-dimensional conformalfield theory, is a sheaf on the configuration space, which is mathematicallyformulated in terms of a vertex operator algebra and its modules. In this paperwe show that the colored parenthesized braid operad acts weakly 2-categoricallyon a module category of a vertex operator algebra whose object is a module $M$such that $M$ and $M^\vee$ are $C_1$-cofinite. If this action induces a1-categorical action, then the module category inherits a balanced braidedtensor category structure, which is shown for a rational $C_2$-cofinite vertexoperator algebra. This gives an alternative proof of a result of Huang andLepowsky.

Vertex operator algebra and colored parenthesized braid operad