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### Data Types as Quotients of Polynomial Functors

Jeremy Avigad, Mario Carneiro, Simon Hudon, Michael Wagner
2019 International Conference on Interactive Theorem Proving
A broad class 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]

Joachim Kock
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

Joachim Kock
2012 Electronical Notes in Theoretical Computer Science
generalisation 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

Claudio Hermida, Bart Jacobs
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

Neil Ghani, Patricia Johann, Clement Fumex, Bartek Klin
2013 Logical Methods in Computer Science
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.  ...  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]

J. H. Davenport, B. M. Trager
1990 Lecture Notes in Computer Science
In this paper we explain how one system solves this problem, what 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]

Clément Fumex, Neil Ghani, Patricia Johann
2011 Lecture Notes in Computer Science
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.  ...  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]

Hao Sun
2020 arXiv   pre-print
In this paper, we define Λ-quot-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]

Nitin Nitsure
2005 arXiv   pre-print
This is an expository account 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]

Mark Blume
2011 arXiv   pre-print
We find a construction 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

Eric Finster, Samuel Mimram, Maxime Lucas, Thomas Seiller
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  ...

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

2011 Sbornik. Mathematics
scheme 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

Hendrik Tews
2004 Electronical Notes in Theoretical Computer Science
In this paper I investigate the case where the 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  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]

Marcelo Fiore
2012 Lecture Notes in Computer Science
The main theoretical contribution is the introduction 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]

Joel Gibson, Lars Thorge Jensen, Geordie Williamson
2022 arXiv   pre-print
bases 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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