TY - JOUR

T1 - What is a categorical model of arrows?

AU - Atkey, Robert

PY - 2011/3/8

Y1 - 2011/3/8

N2 - We investigate what the correct categorical formulation of Hughes’ Arrows should be. It has long been folklore that Arrows, a functional programming construct, and Freyd categories, a categorical notion due to Power, Robinson and Thielecke, are somehow equivalent. In this paper, we show that the situation is more subtle. By considering Arrows wholly within the base category we derive two alternative formulations of Freyd category that are equivalent to Arrows—enriched Freyd categories and indexed Freyd categories. By imposing a further condition, we characterise those indexed Freyd categories that are isomorphic to Freyd categories. The key differentiating point is the number of inputs available to a computation and the structure available on them, where structured input is modelled using comonads.

AB - We investigate what the correct categorical formulation of Hughes’ Arrows should be. It has long been folklore that Arrows, a functional programming construct, and Freyd categories, a categorical notion due to Power, Robinson and Thielecke, are somehow equivalent. In this paper, we show that the situation is more subtle. By considering Arrows wholly within the base category we derive two alternative formulations of Freyd category that are equivalent to Arrows—enriched Freyd categories and indexed Freyd categories. By imposing a further condition, we characterise those indexed Freyd categories that are isomorphic to Freyd categories. The key differentiating point is the number of inputs available to a computation and the structure available on them, where structured input is modelled using comonads.

KW - arrows

KW - categorical model

KW - functional programming

KW - Hughes arrows

KW - Freyd categories

UR - http://www.scopus.com/inward/record.url?scp=79952421374&partnerID=8YFLogxK

UR - http://www.sciencedirect.com/science/article/pii/S157106611100051X

UR - http://personal.cis.strath.ac.uk/~raa/arrows.html

U2 - 10.1016/j.entcs.2011.02.014

DO - 10.1016/j.entcs.2011.02.014

M3 - Conference Contribution

VL - 229

SP - 19

EP - 37

JO - Electronic Notes in Theoretical Computer Science

JF - Electronic Notes in Theoretical Computer Science

SN - 1571-0661

IS - 5

ER -