## Analytic 2-functors

### Mark Weber – 28 May 2003

Using the notion of generic morphism (for an endofunctor) from [1], I described a characterisation of the image of the left kan extension 2-functor
Lan_{I} : [P, Cat] ----> [Cat, Cat]
where P (is a skeletal version of) the category of finite sets and bijections, and I is the (2)-functor that regards each finite set as a discrete category.
This characterisation is a "categorification" of Andre Joyal's characterisation of analytic endofunctors of Set (and the corresponding natural transformations) found in (the appendix of) [2].
It is obtained by viewing endo-2-functors of Cat (in the obvious way) as as endofunctors of the category OpLax[2,Cat] of functors 2 ---> Cat and op-lax natural transformations between them, where 2 is the ordinal {0,1}. By the way, the 2-cells in OpLax[2,Cat] aren't used in this work, so we regard it as a mere category. viewing 2-natural transformations and modifications between endo-2-functors of Cat (both) as natural transformations between the corresponding endofunctors of OpLax[2,Cat]. then applying Joyal's approach to characterising analytic functors (and nats) as formalised in [1].
[1] Mark Weber: Symmetric operads for globular sets. PhD thesis Macquarie University, 2001.
[2] Andre Joyal: Foncteurs analytiques et especes de structures. SLNM 1234 pp 126-159, 1991.

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