We study polynomial functors over locally cartesian closed categories. After setting up the basic theory, we show how polynomial functors assemble into a double category, in fact a framed bicategory. We show that the free monad on a polynomial endofunctor is polynomial. The relationship with operads and other related notions is explored.Comment: 41 pages, latex, 2 ps figures generated at runtime by the texdraw package (does not compile with pdflatex). v2: removed assumptions on sums, added short discussion of generalisation, and more details on tensorial strength

We develop further the theory of weak factorization systems and algebraic weak factorization systems. In particular, we give a method for constructing (algebraic) weak factorization systems whose right maps can be thought of as (uniform) fibrations and that satisfy the (functorial) Frobenius condition. As applications, we obtain a new proof that the Quillen model structure for Kan complexes is right proper, avoiding entirely the use of topological realization and minimal fibrations, and we solve an open problem in the study of Voevodsky's simplicial model of type theory, proving a constructive version of the preservation of Kan fibrations by pushforward along Kan fibrations. Our results also subsume and extend work by Coquand and others on cubical sets. Huber, André Joyal, Andrew Pitts, Emily Riehl, and Andrew Swan for helpful discussions and comments on earlier versions of the paper, to Marco Larrea Schiavon for pointing out an erroneous statement of Proposition 4.2 in an earlier version of this paper, and to Fosco Loregian for pointing us to a useful reference. ⋔ . Of course, a wfs could be free on many different classes of maps. By Quillen's small object argument [34], every set I determines a wfs (L, R) free on I. These are what we call cofibrantly generated wfs's. The next, very simple, proposition provides a convenient way of checking that a cofibrantly generated wfs satisfies the Frobenius condition and will be used in the proof of Theorem 3.8.Proposition 1.4. Let (L, R) be a wfs that is free on a class of maps I. Then the following are equivalent:(i) (L, R) has the Frobenius property. (ii) The pushforward along an R-map preserves R-maps.

Abstract. The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial species considered in the literature-including of course Joyal's original notion-together with their associated substitution operation. Our first main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory for a pseudo-comonad on profunctors. Our second main result establishes that the bicategory of generalised species of structures is cartesian closed.

Abstract. We set out to study the consequences of the assumption of types of wellfounded trees in dependent type theories. We do so by investigating the categorical notion of wellfounded tree introduced in [16]. Our main result shows that wellfounded trees allow us to define initial algebras for a wide class of endofunctors on locally cartesian closed categories.