Partial Evaluations and the Compositional Structure of the Bar Construction release_kfmutj4toneaxmmekktoomivm4

by Carmen Constantin, Paolo Perrone, Tobias Fritz, Brandon Shapiro

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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.
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Date   2020-09-15
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arXiv  2009.07302v1
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