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On generalizing free algebras for a functor

D. C. S. Coumans, S. J. van Gool
2012 Journal of Logic and Computation  
In this paper we introduce a new setting, based on partial algebras, for studying constructions of finitely generated free algebras.  ...  In particular, our method encompasses the construction of finitely generated free algebras for varieties of algebras for a functor as in [2], Heyting algebras as in [1] and S4 algebras as in [8] .  ...  We give a sufficient condition for the free image-total functor for a logic to yield the free total algebra in the colimit. • Section 4.  ... 
doi:10.1093/logcom/exs016 fatcat:rvo2naz4pjd43g5ujid4gobjay

Birings and plethories of integer-valued polynomials

Jesse Elliott
2010 Actes des rencontres du CIRM  
This holds in particular if θn is an isomorphism for all n, which in turn holds, for example, if D is a Krull domain or more generally a TV PVMD.  ...  Let A and B be commutative rings with identity. An A-B-biring is an A-algebra S together with a lift of the functor HomA(S, −) from A-algebras to sets to a functor from A-algebras to B-algebras.  ...  For any set X the ring Int(Z X ) is the free binomial ring generated by X, and a Z-torsion-free ring A is binomial if and only if, for every aA, there exists a ring homomorphism Int(Z) −→ A sending  ... 
doi:10.5802/acirm.34 fatcat:snet665v7je7rkk36mq56t6vla

Comparing free algebras in Topological and Classical Domain Theory

Ingo Battenfeld
2010 Theoretical Computer Science  
Let (Σ, E ) be a parametrized equational theory for kTop. Then the free algebra functor F : kTop → kTop (Σ,E) preserves topological quotient maps.  ...  Moreover if F : kTop → kTop Σ denotes the absolutely free algebra functor, then for any compactly generated space X , the free algebra (FX, {σ FX }) over X is given by (F X / ∼ E , {σ F X /∼ E }), where  ...  I am also very grateful to John Coleman for supplying me with the necessary background on topological algebras, and to Achim Jung for pointing me towards Christine Strüder's thesis and for pointing out  ... 
doi:10.1016/j.tcs.2010.01.021 fatcat:tirndyf2zjfxfj2zafgoaijbje

FUNCTORIAL SEMANTICS OF ALGEBRAIC THEORIES

F. W. Lawvere
1963 Proceedings of the National Academy of Sciences of the United States of America  
useful generality can be achieved; for example, the constructions of free algebras, tensor algebras, monoid rings, enveloping algebras of Lie algebras, abelianization of groups, and covariant extension  ...  Any algebraic theory A is equivalent to the dual of the full category of finitely generated free algebras in its associated algebraic category.  ...  where G is an object in a such that GU4 is a free A-algebra on one generator. These observations enable us to completely characterize algebraic categories in the theorem below.  ... 
doi:10.1073/pnas.50.5.869 pmid:16591125 pmcid:PMC221940 fatcat:meyhvr7pdnax3nbrqqdwe3wxuu

Stable homotopy of algebraic theories

Stefan Schwede
2001 Topology  
For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH }Quillen homology via certain spectral sequences.  ...  For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra.  ...  Acknowledgements This paper is a substantially modi"ed version of the main part of the author's thesis. It could not have been written without many discussions that I had with various colleagues.  ... 
doi:10.1016/s0040-9383(99)00046-4 fatcat:fvifpxxnwzhtxolq62g4ivwhya

A van Kampen spectral sequence for higher homotopy groups

Christopher R. Stover
1990 Topology  
We define the free Idiagram of II-algebras generated by the Ki to be the free diagram 9{F:Ki} where each 9Ki is the free II-algebra on generators Ki as in 4.5.  ...  Naturality of this construction implies that the definition of @ in 7.1 gives a functor on the category of pairs of free II-algebras that is independent of choice of generators.  ...  For the general case let F.A and F.B be free simplicial resolutions of A and B. Let F.A 5 F.B denote the bisimplicial IIalgebra that is F,A 0 F,J3 in dimension p, q.  ... 
doi:10.1016/0040-9383(90)90022-c fatcat:mu3frrz3bzbk7m3sz5p7bsikau

Free objects and Gröbner-Shirshov bases in operated contexts [article]

Zihao Qi, Yufei Qin, Kai Wang, Guodong Zhou
2021 arXiv   pre-print
Various free object functors in these operated contexts are explicitly constructed.  ...  A mild sufficient condition is found such that Φ together with a Gröbner-Shirshov basis of an algebra A form a Gröbner-Shirshov basis of the free Φ-algebra over algebra A in the sense of Guo et al..  ...  Xing Gao, Li Guo and Yunnan Li for inspiring online talks and many useful comments.  ... 
arXiv:2103.13046v1 fatcat:f6zrznd555hwnlksmhhs3si3i4

Differential (Lie) algebras from a functorial point of view

Laurent Poinsot
2015 ACM Communications in Computer Algebra  
This relation is actually functorial, and this functor, as any algebraic functor, is known to admit a left adjoint, namely the universal enveloping algebra of a Lie algebra.  ...  Other standard functorial constructions, such as the tensor and symmetric algebras, are studied for algebras with a given derivation.  ...  This functor admits a left adjoint, known as the free Σ-algebra construction: for a set X, Σ[X] then denotes the free Σ-algebra on X (its Σ-algebra structure is not specified in this notation, and the  ... 
doi:10.1145/2768577.2768634 fatcat:suvmgx33hjcxvpsgnykixlpqta

Differential (Lie) algebras from a functorial point of view

Laurent Poinsot
2016 Advances in Applied Mathematics  
This relation is actually functorial, and this functor, as any algebraic functor, is known to admit a left adjoint, namely the universal enveloping algebra of a Lie algebra.  ...  Other standard functorial constructions, such as the tensor and symmetric algebras, are studied for algebras with a given derivation.  ...  This functor admits a left adjoint, known as the free Σ-algebra construction: for a set X, Σ[X] then denotes the free Σ-algebra on X (its Σ-algebra structure is not specified in this notation, and the  ... 
doi:10.1016/j.aam.2015.09.003 fatcat:bek4kpeu45cklmr2cx6kuajis4

Derived functors of graded algebras

David Blanc
1990 Journal of Pure and Applied Algebra  
A number of spectra1 sequences arising in homotopy theory have the derived functors of a graded algebraic functor as their E2-term.  ...  We also show that under these conditions the nth derived functor, for large n, depends only on low-dimensional information.  ...  Acknowledgment I wish to thank my advisor, Dan Kan, for his help and advice in writing this paper; also Phil Hirschhorn and Haynes Miller for many useful conversations.  ... 
doi:10.1016/0022-4049(90)90060-u fatcat:xgzyw576srg4hfbrq27z2lyska

Homotopy nilpotent groups

Georg Biedermann, William G Dwyer
2010 Algebraic and Geometric Topology  
We prove that the set-valued algebraic theory obtained by applying π_0 is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy  ...  We prove that n-excisive functors of the form Ω F have values in homotopy n-nilpotent groups.  ...  free graded Lie algebra on A.  ... 
doi:10.2140/agt.2010.10.33 fatcat:rmrp4laexrgb7euaoqwflfve7q

Birings and plethories of integer-valued polynomials [article]

Jesse Elliott
2014 arXiv   pre-print
Let A and B be commutative rings with identity. An A-B-biring is an A-algebra S together with a lift of the functor Hom_A(S,-) from A-algebras to sets to a functor from A-algebras to B-algebras.  ...  This holds in particular if θ_n is an isomorphism for all n, which in turn holds, for example, if D is a Krull domain or more generally a TV PVMD.  ...  For any set X the ring Int(Z X ) is the free binomial ring generated by X, and a Z-torsion-free ring A is binomial if and only if, for every aA, there exists a ring homomorphism Int(Z) −→ A sending  ... 
arXiv:1109.3848v3 fatcat:bf67gk6kmrdjzb7bt6fyrqbyl4

A Simple Construction of Derived Representation Schemes [article]

Yuri Berest, George Khachatryan, Ajay Ramadoss
2010 arXiv   pre-print
We present a simple algebraic construction of the (non-abelian) derived functors DRep_n(A) of the representation scheme Rep_n(A), parametrizing the n-dimensional representations of an associative algebra  ...  We construct a related derived version of the representation functor introduced recently by M. Van den Bergh and, as an application, compute the derived tangent spaces TDRep_n(A) to Rep_n(A).  ...  We are also very grateful to Martin Kassabov for several insightful suggestions and to Frank Moore, whose computer DG algebra package for Macaulay2 we have been using extensively.  ... 
arXiv:1010.4901v1 fatcat:7lioapqdnbbzloeejyvitqha7y

An intrinsic definition of the Rees algebra of a module [article]

Gustav Sædén Ståhl
2014 arXiv   pre-print
This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring.  ...  Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.  ...  the module into a certain free module and taking the image of the map given by applying the symmetric algebra functor on this embedding.  ... 
arXiv:1402.3219v2 fatcat:nj4whu3pnvghthcl32r7e6ofl4

PBW property for associative universal enveloping algebras over an operad [article]

Anton Khoroshkin
2020 arXiv   pre-print
A necessary condition on the Hilbert series for P is discovered.  ...  We study the notion of PBW property for universal enveloping algebras over an operad. In case P is Koszul a criterion for the PBW property is found.  ...  Proposition 4 . 21 . 421 The functors Sh and + commute and, moreover, they map free objects to free objectsIn particular, the functors Sh preserve the number of generators of a free symmetric operad and  ... 
arXiv:1807.05873v3 fatcat:gms4bp43xnhb7km773bq42vztu
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