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Although it has been a well-known fact, for more than two decades, thatcategory theory is needed for the study of topological orders, it is still anon-trivial challenge for students and working physicists to master theabstract language of category theory. In this work, for those who have nobackground in category theory, we explain in great details how the structure ofa (braided) fusion category naturally emerges from lattice models and physicalintuitions. Moreover, we show that nearly all mathematical notions andconstructions in fusion categories and its representation theory, such as(monoidal) functors, Drinfeld center, module categories, Morita equivalence,condensation completion and fusion 2-categories, naturally emerge from latticemodels and physical intuitions. In this process, we also introduce some basicnotions and important results of topological orders.

An invitation to topological orders and category theory