Displayed categories provide a natural categorification from classifying functions to the world of functors. The spirit of the idea is to encode a functor $ F: D \to C $ using a suitable 2-functor (lax double functor, lax pseudofunctor, depending on context) $ F^{-1} : C \to Span $ into the suitable 2-category (double category, bicategory) of spans of functions of sets. If we iterate this process again and construct a display functor $ F^{{-1}^{-1}} $ for $ F^{-1} $ and obtain some kind of 2-endofunctor $ Span\to Span $, have we effectively encoded 1-categorical data into the suitably-monoidal 2-category of 2-endofunctors on $ Span $? Does this provide an embedding of the 2-category $ Cat $ of small categories into a monoidal 2-category?
$\begingroup$
$\endgroup$
1
-
3$\begingroup$ How do you propose to iterate this construction? Its input is an ordinary functor, but its output is a lax functor. $\endgroup$– Alexander CampbellCommented Nov 18, 2019 at 10:28
Add a comment
|