5,807 research outputs found
Two applications of analytic functors
AbstractWe apply the theory of analytic functors to two topics related to theoretical computer science. One is a mathematical foundation of certain syntactic well-quasi-orders and well-orders appearing in graph theory, the theory of term rewriting systems, and proof theory. The other is a new verification of the Lagrange–Good inversion formula using several ideas appearing in semantics of lambda calculi, especially the relation between categorical traces and fixpoint operators
Comparing the orthogonal and homotopy functor calculi
Goodwillie's homotopy functor calculus constructs a Taylor tower of
approximations to F, often a functor from spaces to spaces. Weiss's orthogonal
calculus provides a Taylor tower for functors from vector spaces to spaces. In
particular, there is a Weiss tower associated to the functor which sends a
vector space V to F evaluated at the one-point compactification of V.
In this paper, we give a comparison of these two towers and show that when F
is analytic the towers agree up to weak equivalence. We include two main
applications, one of which gives as a corollary the convergence of the Weiss
Taylor tower of BO. We also lift the homotopy level tower comparison to a
commutative diagram of Quillen functors, relating model categories for
Goodwillie calculus and model categories for the orthogonal calculus.Comment: 28 pages, sequel to Capturing Goodwillie's Derivative,
arXiv:1406.042
Analytic geometry over F_1 and the Fargues-Fontaine curve
This paper develops a theory of analytic geometry over the field with one
element. The approach used is the analytic counter-part of the Toen-Vaquie
theory of schemes over F_1, i.e. the base category relative to which we work
out our theory is the category of sets endowed with norms (or families of
norms). Base change functors to analytic spaces over Banach rings are studied
and the basic spaces of analytic geometry (like polydisks) are recovered as a
base change of analytic spaces over F_1. We end by discussing some applications
of our theory to the theory of the Fargues-Fontaine curve and to the ring Witt
vectors.Comment: Small corrections have been made in the last section of the paper and
some typos have been correcte
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
Shapely monads and analytic functors
In this paper, we give precise mathematical form to the idea of a structure
whose data and axioms are faithfully represented by a graphical calculus; some
prominent examples are operads, polycategories, properads, and PROPs. Building
on the established presentation of such structures as algebras for monads on
presheaf categories, we describe a characteristic property of the associated
monads---the shapeliness of the title---which says that "any two operations of
the same shape agree". An important part of this work is the study of analytic
functors between presheaf categories, which are a common generalisation of
Joyal's analytic endofunctors on sets and of the parametric right adjoint
functors on presheaf categories introduced by Diers and studied by
Carboni--Johnstone, Leinster and Weber. Our shapely monads will be found among
the analytic endofunctors, and may be characterised as the submonads of a
universal analytic monad with "exactly one operation of each shape". In fact,
shapeliness also gives a way to define the data and axioms of a structure
directly from its graphical calculus, by generating a free shapely monad on the
basic operations of the calculus. In this paper we do this for some of the
examples listed above; in future work, we intend to do so for graphical calculi
such as Milner's bigraphs, Lafont's interaction nets, or Girard's
multiplicative proof nets, thereby obtaining canonical notions of denotational
model
Polynomial Functors of Modules
We introduce the notion of numerical functors to generalise Eilenberg &
MacLane's polynomial functors to modules over a binomial base ring. After
shewing how these functors are encoded by modules over a certain ring, we
record a precise criterion for a numerical (or polynomial) functor to admit a
strict polynomial structure in the sense of Friedlander & Suslin. We also
provide several characterisations of analytic functors
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