For accessible set-valued functors it is well known that weak preservation of limits is equivalent to representability, and weak preservation of connected limits to familial representability. In contrast, preservation of weak wide pullbacks is equivalent to being a coproduct of quotients of $\hom$-functors modulo groups of automorphisms. For finitary functors this was proved by Andr\'e Joyal who called these functors analytic. We introduce a generalization of Joyal's concept from endofunctors of Set to endofunctors of a symmetric monoidal category.
Keywords: analytic functor, weak limit, weak pullback
2000 MSC: 18A25, 18D10, 18B05
Theory and Applications of Categories,
Vol. 21, 2008,
No. 11, pp 191-209.