## An op/lax Fam construction

### Jason Brown – 20 July 2022

The free completion of a category C under coproducts is given by the 'Fam construction', Fam(C). One dimension higher, one can consider the free completion of a 2-category under colimits of functors from 1-categories. When these functors and colimits are allowed to be lax/oplax the result is something similar to the Fam construction. We might call it the 'op/lax Fam construction'. The op/lax Fam construction also bears similarity to the co/Kleisli completion, which it contains as a full subcategory. I'll describe this construction, mention some interesting properties, and give examples.

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