Partial Evaluations and the Compositional Structure of the Bar Construction
release_kfmutj4toneaxmmekktoomivm4
by
Carmen Constantin, Paolo Perrone, Tobias Fritz, Brandon Shapiro
2020
Abstract
An algebraic expression like 3 + 2 + 6 can be evaluated to 11, but it can
also be partially evaluated to 5 + 6. In categorical algebra, such
partial evaluations can be defined in terms of the 1-skeleton of the bar
construction for algebras of a monad. We show that this partial evaluation
relation can be seen as the relation internal to the category of algebras
generated by relating a formal expression to its result. The relation is
transitive for many monads which describe commonly encountered algebraic
structures, and more generally for BC monads on , defined by the
underlying functor and multiplication being weakly cartesian. We find that this
is not true for all monads: we describe a finitary monad on for which
the partial evaluation relation on the terminal algebra is not transitive.
With the perspective of higher algebraic rewriting in mind, we then
investigate the compositional structure of the bar construction in all
dimensions. We show that for algebras of BC monads, the bar construction has
fillers for all directed acyclic configurations in Δ^n, but
generally not all inner horns. We introduce several additional
completeness and exactness conditions on simplicial sets which
correspond via the bar construction to composition and invertibility properties
of partial evaluations, including those arising from weakly cartesian
monads. We characterize and produce factorizations of pushouts and certain
commutative squares in the simplex category in order to provide simplified
presentations of these conditions and relate them to more familiar properties
of simplicial sets.
In text/plain
format
Archived Files and Locations
application/pdf 807.9 kB
file_62o7l5xqpfeojhmfrjdeiqreca
|
arxiv.org (repository) web.archive.org (webarchive) |
2009.07302v1
access all versions, variants, and formats of this works (eg, pre-prints)