618 research outputs found
Homological Computations for Term Rewriting Systems
An important problem in universal algebra consists in finding
presentations of algebraic theories by generators and relations, which
are as small as possible. Exhibiting lower bounds on the number of
those generators and relations for a given theory is a difficult task
because it a priori requires considering all possible sets of
generators for a theory and no general method exists. In this article,
we explain how homological computations can provide such lower bounds,
in a systematic way, and show how to actually compute those in the
case where a presentation of the theory by a convergent rewriting
system is known. We also introduce the notion of coherent presentation
of a theory in order to consider finer homotopical invariants. In some
aspects, this work generalizes, to term rewriting systems, Squier\u27s
celebrated homological and homotopical invariants for string rewriting
systems
Higher-dimensional normalisation strategies for acyclicity
We introduce acyclic polygraphs, a notion of complete categorical cellular
model for (small) categories, containing generators, relations and
higher-dimensional globular syzygies. We give a rewriting method to construct
explicit acyclic polygraphs from convergent presentations. For that, we
introduce higher-dimensional normalisation strategies, defined as homotopically
coherent ways to relate each cell of a polygraph to its normal form, then we
prove that acyclicity is equivalent to the existence of a normalisation
strategy. Using acyclic polygraphs, we define a higher-dimensional homotopical
finiteness condition for higher categories which extends Squier's finite
derivation type for monoids. We relate this homotopical property to a new
homological finiteness condition that we introduce here.Comment: Final versio
Free resolutions via Gr\"obner bases
For associative algebras in many different categories, it is possible to
develop the machinery of Gr\"obner bases. A Gr\"obner basis of defining
relations for an algebra of such a category provides a "monomial replacement"
of this algebra. The main goal of this article is to demonstrate how this
machinery can be used for the purposes of homological algebra. More precisely,
our approach goes in three steps. First, we define a combinatorial resolution
for the monomial replacement of an object. Second, we extract from those
resolutions explicit representatives for homological classes. Finally, we
explain how to "deform" the differential to handle the general case. For
associative algebras, we recover a well known construction due to Anick. The
other case we discuss in detail is that of operads, where we discover
resolutions that haven't been known previously. We present various
applications, including a proofs of Hoffbeck's PBW criterion, a proof of
Koszulness for a class of operads coming from commutative algebras, and a
homology computation for the operads of Batalin--Vilkovisky algebras and of
Rota--Baxter algebras.Comment: 34 pages, 4 figures. v2: added references to the work of
Drummond-Cole and Vallette. v3: added an explicit description of homology
classes in the monomial case and more examples, re-structured the exposition
to achieve more clarity. v4: changed the presentation of the main
construction to make it clearer, added another example (a computation of the
bar homology of Rota--Baxter algebras
Quillen homology for operads via Gr\"obner bases
The main goal of this paper is to present a way to compute Quillen homology
of operads. The key idea is to use the notion of a shuffle operad we introduced
earlier; this allows to compute, for a symmetric operad, the homology classes
and the shape of the differential in its minimal model, although does not give
an insight on the symmetric groups action on the homology. Our approach goes in
several steps. First, we regard our symmetric operad as a shuffle operad, which
allows to compute its Gr\"obner basis. Next, we define a combinatorial
resolution for the "monomial replacement" of each shuffle operad (provided by
the Gr\"obner bases theory). Finally, we explain how to "deform" the
differential to handle every operad with a Gr\"obner basis, and find explicit
representatives of Quillen homology classes for a large class of operads. We
also present various applications, including a new proof of Hoffbeck's PBW
criterion, a proof of Koszulness for a class of operads coming from commutative
algebras, and a homology computation for the operads of Batalin-Vilkovisky
algebras and of Rota-Baxter algebras.Comment: 41 pages, this paper supersedes our previous preprint
arXiv:0912.4895. Final version, to appear in Documenta Mat
Syzygies among reduction operators
We introduce the notion of syzygy for a set of reduction operators and relate
it to the notion of syzygy for presentations of algebras. We give a method for
constructing a linear basis of the space of syzygies for a set of reduction
operators. We interpret these syzygies in terms of the confluence property from
rewriting theory. This enables us to optimise the completion procedure for
reduction operators based on a criterion for detecting useless reductions. We
illustrate this criterion with an example of construction of commutative
Gr{\"o}bner basis
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