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This paper clarifies basic definitions in the universal construction oftopological theories and monoidal categories. The definition of the universalconstruction is given for various types of monoidal categories, including rigidand symmetric. It is also explained how to set up the universal constructionfor non-monoidal categories. The second part of the paper explains how toassociate a rigid symmetric monoidal category to a small category, a sort ofthe Brauer envelope of a category. The universal construction for the Brauerenvelopes generalizes some earlier work of the first two authors on automata,power series and topological theories. Finally, the theory of pseudocharacters(or pseudo-representations), which is an essential tool in modern numbertheory, is interpreted via one-dimensional topological theories and TQFTs withdefects. The notion of a pseudocharacter is studied for Brauer categories andthe lifting property to characters of semisimple representations is establishedin characteristic 0 for Brauer categories with at most countably many objects.The paper contains a brief discussion of pseudo-holonomies, which are functionsfrom loops in a manifold to real numbers similar to traces of the holonomiesalong loops of a connection on a vector bundle on the manifold. It concludeswith a classification of pseudocharacters (pseudo-TQFTs) and their generatingfunctions for the category of oriented two-dimensional cobordisms in thecharacteristic 0 case.

Universal construction in monoidal and non-monoidal settings, the Brauer envelope, and pseudocharacters