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Data Types as Quotients of Polynomial Functors

2019
*
International Conference on Interactive Theorem Proving
*

A broad class

doi:10.4230/lipics.itp.2019.6
dblp:conf/itp/AvigadCH19
fatcat:hf6sw36i5bfmppfdme3upn6se4
*of**data**types*, including arbitrary nestings*of*inductive*types*, coinductive*types*, and*quotients*, can be represented*as**quotients**of**polynomial**functors*. ... This provides perspicuous ways*of*constructing them and reasoning about them in an interactive theorem prover. ... I T P 2 0 1 9 6:4*Data**Types**as**Quotients**of**Polynomial**Functors*We have found that working with QPFs is natural and intuitive. ...##
###
Data types with symmetries and polynomial functors over groupoids
[article]

2012
*
arXiv
*
pre-print

*Polynomial*

*functors*are useful in the theory

*of*

*data*

*types*, where they are often called containers. ... 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. ... The

*data*to be inserted into these positions can be

*of*any

*type*X: the

*polynomial*

*functor*receives a

*type*X (a set) and returns the new more elaborate

*type*X E b . ...

##
###
Data Types with Symmetries and Polynomial Functors over Groupoids

2012
*
Electronical Notes in Theoretical Computer Science
*

generalisation

doi:10.1016/j.entcs.2013.01.001
fatcat:kj3wplor6jhdvjwdlv245ckcla
*of*and a clean unifying framework for*quotient*containers (in the sense*of*Abbott et al.), species and analytic*functors*(Joyal 1985),*as*well*as*the stuff*types**of*Baez and Dolan. ...*Polynomial**functors*(over Set or other locally cartesian closed categories) are useful in the theory*of**data**types*, where they are often called containers. ... The*data*to be inserted into these positions can be*of*any*type*X: the*polynomial**functor*receives a*type*X (a set) and returns the new more elaborate*type*X E b . ...##
###
Structural Induction and Coinduction in a Fibrational Setting

1998
*
Information and Computation
*

*of*

*quotient*

*types*, the coinduction principle for terminal coalgebras is admissible. ... We present a categorical logic formulation

*of*induction and coinduction principles for reasoning about inductively and coinductively de ned

*types*. ... Dually, coinductive

*data*

*types*correspond to terminal coalgebras (C; C d ! TC), where T speci es the signature

*of*destructors

*of*the

*data*

*type*and d maps an element

*of*the

*type*C to its components. ...

##
###
Indexed Induction and Coinduction, Fibrationally

2013
*
Logical Methods in Computer Science
*

That is, we present a sound coinduction rule for any

doi:10.2168/lmcs-9(3:6)2013
fatcat:77pcj77n5zfshpvcwstsrvx5zq
*data**type*arising*as*the carrier*of*the final coalgebra*of*a*functor*, thus relaxing Hermida and Jacobs' restriction to*polynomial**functors*. ... Indexed*data**types*often arise*as*carriers*of*initial algebras and final coalgebras*of**functors*on slice categories, so we give sufficient conditions under which we can construct, from a CCU (QCE) U:E ... But this theory, too, is sound only for*polynomial**data**types*, and so does not apply to final coalgebras*of*some key*functors*, such*as*the finite powerset*functor*. ...##
###
Scratchpad's view of algebra I: Basic commutative algebra
[chapter]

1990
*
Lecture Notes in Computer Science
*

In this paper we explain how one system solves this problem, what

doi:10.1007/3-540-52531-9_122
fatcat:jnhob34v4bg2tnysmq7e6z27ge
*types*and operators it is necessary to introduce and, in short, how one can construct a computational theory*of*commutative algebra. ... While computer algebra systems have dealt with*polynomials*and rational functions with integer coefficients for many years, dealing with more general constructs from commutative algebra is a more recent ... Much discussion*of*this material took place while the authors enjoyed the hospitality*of*Mrs. Barbara Gatje. ...##
###
Indexed Induction and Coinduction, Fibrationally
[chapter]

2011
*
Lecture Notes in Computer Science
*

That is, we present a sound coinduction rule for any

doi:10.1007/978-3-642-22944-2_13
fatcat:lons4fr3xnfndmrzhi7roncwaa
*data**type*arising*as*the carrier*of*the final coalgebra*of*a*functor*, thus relaxing Hermida and Jacobs' restriction to*polynomial**functors*. ... That is, we present a sound coinduction rule for any*data**type*arising*as*the carrier*of*the final coalgebra*of*a*functor*, thus relaxing Hermida and Jacobs' restriction to*polynomial**functors*. ... But this theory, too, is sound only for*polynomial**data**types*, and so does not apply to final coalgebras*of*some key*functors*, such*as*the finite powerset*functor*. ...##
###
Moduli Space of Λ-modules on Projective Deligne-Mumford Stacks
[article]

2020
*
arXiv
*
pre-print

In this paper, we define Λ-quot-

arXiv:2003.11674v2
fatcat:ssnp2xz3tbawfa7rev45ekad4a
*functors*on Deligne-Mumford stacks. We prove that the Λ-quot-*functor*is representable by an algebraic space. ... Then, we construct the moduli space*of*Λ-modules on a projective Deligne-Mumford stack. We prove that this moduli space is a quasi-projective scheme. ... Let P be an integer*polynomial*(*as*modified Hilbert*polynomial*), and d is the degree*of*P , which is a positive integer (*as*pure dimension). ...##
###
Construction of Hilbert and Quot Schemes
[article]

2005
*
arXiv
*
pre-print

This is an expository account

arXiv:math/0504590v1
fatcat:r3w46tgwo5dtzp324rrs3yge6i
*of*Grothendieck's construction*of*Hilbert and Quot Schemes, following his talk 'Techniques de construction et theoremes d'existence en geometrie algebriques IV : les schemas ... These notes are based on a series*of*six lectures in the summer school 'Advanced Basic Algebraic Geometry', held at the Abdus Salam International Centre for Theoretical Physics, Trieste, in July 2003. ...*As*the parameter scheme X is noetherian and*as*the Hilbert*polynomial*is locally constant, only finitely many*polynomials*Φ occur*as*Hilbert*polynomials**of*fibers*of*f 2 : R → X with respect to a chosen ...##
###
Construction of G-Hilbert schemes
[article]

2011
*
arXiv
*
pre-print

We find a construction

arXiv:math/0607577v4
fatcat:irb6jqtmizh27ifwyslecpmsru
*of*G-Hilbert schemes*as*relative G-Hilbert schemes over the*quotient*that does not need the Hilbert scheme*of*n points, works under more natural assumptions and gives additional ... information about the morphism from the G-Hilbert scheme to the*quotient*. ... One may rewrite the corresponding*functor**as*3. Let f : X S be locally*of*finite*type*over the noetherian scheme S. ...##
###
A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory

2021
*
Electronic Proceedings in Theoretical Computer Science
*

*Polynomial*

*functors*are a categorical generalization

*of*the usual notion

*of*

*polynomial*, which has found many applications in higher categories and

*type*theory: those are generated by

*polynomials*consisting ... Notably, this requires us introducing an axiomatization in a small universe

*of*the

*type*

*of*finite

*types*,

*as*an appropriate higher inductive

*type*

*of*natural numbers and bijections. ... Writing Set/I for the slice category

*of*sets over a set I, such

*data*again induces a

*functor*P : Set/I → Set/J, called a

*polynomial*

*functor*, obtained

*as*the composite P = Σ t • Π p • ∆ s where ∆ s is the ...

##
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On a new compactification of moduli of vector bundles on a surface. III: Functorial approach

2011
*
Sbornik. Mathematics
*

scheme

doi:10.1070/sm2011v202n03abeh004151
fatcat:amygf77sbffdnbzfhl7bcil7ei
*of*semistable vector bundles. ... The construction for nonreduced projective moduli scheme*of*semistable admissible pairs is performed. ... There exists GIT-*quotient*M = H 0 //P GL(V ) which is moduli scheme for the*functor*f. The scheme M is a projective algebraic Noetherian scheme*of*finite*type*. ...##
###
Predicate and Relation Lifting for Parametric Algebraic Specifications

2004
*
Electronical Notes in Theoretical Computer Science
*

In this paper I investigate the case where the

doi:10.1016/j.entcs.2004.02.040
fatcat:b3z4gqbxvvdwvkje32t6k4q52e
*functor*F is defined*as*the initial semantics*of*a (single sorted) parametric algebraic specification. Then: Proof. ... The related notion*of*predicate lifting can be used to define invariants for F -coalgebras. Predicate and relation lifting can be directly defined for a rich class*of**polynomial**functors*[5, 6, 19] . ... In [5] Hensel and Jacobs investigate the case*of*iterated*data**types*, that is, where all these specification do not contain any axioms. ...##
###
Discrete Generalised Polynomial Functors
[chapter]

2012
*
Lecture Notes in Computer Science
*

The main theoretical contribution is the introduction

doi:10.1007/978-3-642-31585-5_22
fatcat:uxt3ecyrbvejdiazdgdfedviru
*of*discrete generalised*polynomial**functors*, a class that lies in between the classes*of*cocontinuous and finitary*functors*, and is closed under composition ... We study generalised*polynomial**functors*between presheaf categories, developing their mathematical theory together with computational applications. ... In modern theories*of**data*structure there has been a need for the naive notion*of**polynomial**functor**as*a sum-*of*-products construction to evolve to more sophisticated ones. ...##
###
Calculating the p-canonical basis of Hecke algebras
[article]

2022
*
arXiv
*
pre-print

bases

arXiv:2204.04924v2
fatcat:3nwm4fngxvg5vkxxfxfkwsvmei
*of*their morphism spaces. ... We describe an algorithm for computing the p-canonical basis*of*the Hecke algebra, or one*of*its antispherical modules. ... Note that this*functor*is not monoidal,*as*the*quotient*category is not monoidal. ...
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