The classifying functor from categories to topological spaces provides a way of constructing

spaces with certain properties or structure from categories with similar properties of structure.

An important example of this is the construction of infinite loop spaces from symmetric monoidal

categories. The particular kinds of extra structure can typically be encoded by monads on the

category of small categories. In order to provide more flexibility in the kinds of morphisms allowed,

one can work with the associated 2-monad in the 2-category of categories, functors, and natural

transformations. In this talk I will give the categorical setup required, and I will give examples of

interest to homotopy theorists. I will also outline how this method of working can give general

statements about strictifications and comparisons of homotopy theories. This is partially based on

work with two different sets of collaborators: Nick Gurski, Niles Johnson, and Marc Stephan;

Bert Guillou, Peter May, and Mona Merling.

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