Local fibered right adjoints are polynomial

For any locally cartesian closed category E, we prove that a local fibered right adjoint between slices of E is given by a polynomial. The slices in question are taken in a well known fibered sense.Comment: 15 pages, LaTeX; final version (v2), to appear in Math. Struct. Comp. Sc

Characteristic numbers of rational curves with cusp or prescribed triple contact

This note pursues the techniques of modified psi classes on the stack of stable maps (cf. [Graber-Kock-Pandharipande]) to give concise solutions to the characteristic number problem of rational curves in P^2 or P^1 x P^1 with a cusp or a prescribed triple contact. The classes of such loci are computed in terms of modified psi classes, diagonal classes, and certain codimension-2 boundary classes. Via topological recursions the generating functions for the numbers can then be expressed in terms of the usual characteristic number potentials.Comment: 19 pages, LaTeX, uses rsfs fonts. Tables and maple code hidden in sourc

Some matrices with nilpotent entries, and their determinants

We study algebraic properties of matrices whose rows are mutual neighbours, and are also neigbours of 0 ("neighbour" in the sense of a certain nilpotency condition). The intended application is in synthetic differential geometry. For a square matrix of this kind, the product of the diagonal entries equals the determinant, modulo a factor n

Theory of characteristics for first order partial differential equations

We use the method of synthetic differential geometry to revisit the geometric reasoning employed by Lie, Klein and others in their study of partial differential equations

Elementary remarks on units in monoidal categories

We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellative idempotent (in a 1-categorical sense). This notion is more economical than the usual notion in terms of left-right constraints, and is motivated by higher category theory. To start, we describe the semi-monoidal category of all possible unit structures on a given semi-monoidal category and observe that it is contractible (if nonempty). Then we prove that the two notions of units are equivalent in a strong functorial sense. Next, it is shown that the unit compatibility condition for a (strong) monoidal functor is precisely the condition for the functor to lift to the categories of units, and it is explained how the notion of Saavedra unit naturally leads to the equivalent non-algebraic notion of fair monoidal category, where the contractible multitude of units is considered as a whole instead of choosing one unit. To finish, the lax version of the unit comparison is considered. The paper is self-contained. All arguments are elementary, some of them of a certain beauty.Comment: Comments: LaTeX, 29 pages. Uses Paul Taylor's diagrams package. Does not compile properly with pdflatex. Version v2: introduction rewritten, minor expository improvements (no mathematical changes), submitted for publicatio

Perturbative renormalisation for not-quite-connected bialgebras

We observe that the Connes--Kreimer Hopf-algebraic approach to perturbative renormalisation works not just for Hopf algebras but more generally for filtered bialgebras $B$ with the property that $B_0$ is spanned by group-like elements (e.g. pointed bialgebras with the coradical filtration). Such bialgebras occur naturally both in Quantum Field Theory, where they have some attractive features, and elsewhere in Combinatorics, where they cover a comprehensive class of incidence bialgebras. In particular, the setting allows us to interpret M\"obius inversion as an instance of renormalisation.Comment: 12 pages, comments are most welcom

Recursion for twisted descendants and characteristic numbers of rational curves

On a space of stable maps, the psi classes are modified by subtracting certain boundary divisors. The top products of modified psi classes, usual psi classes, and classes pulled back along the evaluation maps are called twisted descendants; it is shown that in genus 0, they admit a complete recursion and are determined by the Gromov-Witten invariants. One motivation for this construction is that all characteristic numbers (of rational curves) can be interpreted as twisted descendants; this is explained in the second part, using pointed tangency classes. As an example, some of Schubert's numbers of twisted cubics are verified.Comment: 20 pages, LaTeX, uses Paul Taylor's commutative diagrams packag

Bundle functors and fibrations

We give an account, in terms of fibered categories and their fibrewise duals, of aspects of the theory of bundle functors and star-bundle functors in differential geometry.Comment: 24 page

Graphs, hypergraphs, and properads

A categorical formalism for directed graphs is introduced, featuring natural notions of morphisms and subgraphs, and leading to two elementary descriptions of the free-properad monad, first in terms of presheaves on elementary graphs, second in terms of groupoid-enriched hypergraphs.Comment: v2: substantial revision: corrected a few mistakes concerning convexity; added more details regarding colimits of graphs, associativity of the monad multiplication, generic factorisation, and the nerve theorem; added six references; 40 pages. v3: typos and references; submitte

Affine combinations in affine schemes

We prove that finite sets of mutual neighbor points in an affine scheme admit affine combinations, preserved by any map. Furthermore, such combination has a value which is neighbor point of all the original points