## The Maximality of Cartesian Categories (1999)

### BibTeX

@MISC{99themaximality,

author = {},

title = {The Maximality of Cartesian Categories},

year = {1999}

}

### OpenURL

### Abstract

It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categories, which is related to model-theoretical methods of normalization. The equations between arrows assumed for cartesian categories are maximal in the sense that extending them with new equations collapses the categories into preorders (i.e. categories in which between any two objects there is at most one arrow). The equations envisaged for the extension are in the language of free cartesian categories generated by sets of objects, and variables for arrows don’t occur in them. If such an equation doesn’t hold in the free cartesian category generated by a set of objects, then any cartesian category in which this equation holds is a preorder. An analogous result is provable for categories with binary products, which differ from cartesian categories in not necessarily having a terminal object. The proof of these results, which we are going to present below, is based on a coherence property of cartesian categories. This coherence, which is ultimately inspired by the geometric modelling of categories of [3], is related to model-theoretic methods of normalization. It permits to establish uniqueness of normal form for arrow terms without proceeding via the Church-Rosser property for reductions. It also yields an easy decision procedure for the commuting of diagrams in free cartesian categories.

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