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The algebraic expression 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 total evaluation. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on Set (which are those monads for which the underlying functor and the multiplication are weakly cartesian). We find that this is not true for all monads: we describe a finitary monad on Set 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.
@article{TAC_2023_39_a10,
author = {Carmen Constantin and Tobias Fritz and Paolo Perrone and Brandon T. Shapiro},
title = {Partial evaluations and the compositional structure of the bar construction},
journal = {Theory and applications of categories},
pages = {322--364},
publisher = {mathdoc},
volume = {39},
year = {2023},
language = {en},
url = {http://geodesic.mathdoc.fr/item/TAC_2023_39_a10/}
}
TY - JOUR AU - Carmen Constantin AU - Tobias Fritz AU - Paolo Perrone AU - Brandon T. Shapiro TI - Partial evaluations and the compositional structure of the bar construction JO - Theory and applications of categories PY - 2023 SP - 322 EP - 364 VL - 39 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/TAC_2023_39_a10/ LA - en ID - TAC_2023_39_a10 ER -
%0 Journal Article %A Carmen Constantin %A Tobias Fritz %A Paolo Perrone %A Brandon T. Shapiro %T Partial evaluations and the compositional structure of the bar construction %J Theory and applications of categories %D 2023 %P 322-364 %V 39 %I mathdoc %U http://geodesic.mathdoc.fr/item/TAC_2023_39_a10/ %G en %F TAC_2023_39_a10
Carmen Constantin; Tobias Fritz; Paolo Perrone; Brandon T. Shapiro. Partial evaluations and the compositional structure of the bar construction. Theory and applications of categories, Tome 39 (2023), pp. 322-364. http://geodesic.mathdoc.fr/item/TAC_2023_39_a10/