3,181 research outputs found
Rational Combinatorics
We propose a categorical setting for the study of the combinatorics of
rational numbers. We find combinatorial interpretation for the Bernoulli and
Euler numbers and polynomials.Comment: Adv. in Appl. Math. (2007), doi:10.1016/j.aam.2006.12.00
Data types with symmetries and polynomial functors over groupoids
Polynomial functors are useful in the theory of data types, where they are
often called containers. They are also useful in algebra, combinatorics,
topology, and higher category theory, and in this broader perspective the
polynomial aspect is often prominent and justifies the terminology. For
example, Tambara's theorem states that the category of finite polynomial
functors is the Lawvere theory for commutative semirings. In this talk I will
explain how an upgrade of the theory from sets to groupoids is useful to deal
with data types with symmetries, and provides a common generalisation of and a
clean unifying framework for quotient containers (cf. Abbott et al.), species
and analytic functors (Joyal 1985), as well as the stuff types of Baez-Dolan.
The multi-variate setting also includes relations and spans, multispans, and
stuff operators. An attractive feature of this theory is that with the correct
homotopical approach - homotopy slices, homotopy pullbacks, homotopy colimits,
etc. - the groupoid case looks exactly like the set case. After some standard
examples, I will illustrate the notion of data-types-with-symmetries with
examples from quantum field theory, where the symmetries of complicated tree
structures of graphs play a crucial role, and can be handled elegantly using
polynomial functors over groupoids. (These examples, although beyond species,
are purely combinatorial and can be appreciated without background in quantum
field theory.) Locally cartesian closed 2-categories provide semantics for
2-truncated intensional type theory. For a fullfledged type theory, locally
cartesian closed \infty-categories seem to be needed. The theory of these is
being developed by D.Gepner and the author as a setting for homotopical
species, and several of the results exposed in this talk are just truncations
of \infty-results obtained in joint work with Gepner. Details will appear
elsewhere.Comment: This is the final version of my conference paper presented at the
28th Conference on the Mathematical Foundations of Programming Semantics
(Bath, June 2012); to appear in the Electronic Notes in Theoretical Computer
Science. 16p
Polynomial functors and combinatorial Dyson-Schwinger equations
We present a general abstract framework for combinatorial Dyson-Schwinger
equations, in which combinatorial identities are lifted to explicit bijections
of sets, and more generally equivalences of groupoids. Key features of
combinatorial Dyson-Schwinger equations are revealed to follow from general
categorical constructions and universal properties. Rather than beginning with
an equation inside a given Hopf algebra and referring to given Hochschild
-cocycles, our starting point is an abstract fixpoint equation in groupoids,
shown canonically to generate all the algebraic structure. Precisely, for any
finitary polynomial endofunctor defined over groupoids, the system of
combinatorial Dyson-Schwinger equations has a universal solution,
namely the groupoid of -trees. The isoclasses of -trees generate
naturally a Connes-Kreimer-like bialgebra, in which the abstract
Dyson-Schwinger equation can be internalised in terms of canonical
-operators. The solution to this equation is a series (the Green function)
which always enjoys a Fa\`a di Bruno formula, and hence generates a
sub-bialgebra isomorphic to the Fa\`a di Bruno bialgebra. Varying yields
different bialgebras, and cartesian natural transformations between various
yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to
truncation of Dyson-Schwinger equations. Finally, all constructions can be
pushed inside the classical Connes-Kreimer Hopf algebra of trees by the
operation of taking core of -trees. A byproduct of the theory is an
interpretation of combinatorial Green functions as inductive data types in the
sense of Martin-L\"of Type Theory (expounded elsewhere).Comment: v4: minor adjustments, 49pp, final version to appear in J. Math. Phy
Fundamental groupoids of k-graphs
k-graphs are higher-rank analogues of directed graphs which were first
developed to provide combinatorial models for operator algebras of
Cuntz-Krieger type. Here we develop a theory of the fundamental groupoid of a
k-graph, and relate it to the fundamental groupoid of an associated graph
called the 1-skeleton. We also explore the failure, in general, of k-graphs to
faithfully embed into their fundamental groupoids.Comment: 12 page
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