Enriched and monoidal categories

A monoidal category $\mathcal V$ has a tensor bifunctor, a unit object I, and natural associator and left and right unit isomorphisms. The associator satisfies a pentagon identity, and the unit maps satisfy a triangle identity. These coherence conditions ensure that differently bracketed compositions interact consistently. Associativity is not assumed to be literal equality of objects: even Cartesian products are ordinarily connected by canonical rebracketing isomorphisms rather than identical ordered-pair encodings.

A symmetric monoidal category has additional coherent symmetry maps. A closed monoidal category has internal Hom objects implementing tensor–Hom adjunctions. Neither symmetry nor closedness is required merely to define an enriched category over a monoidal base. They become necessary assumptions for particular further constructions and theorems. For example, a general enriched functor category may demand suitable closedness and limits; those properties cannot be inferred just from the existence of a tensor.

A $\mathcal V$-enriched category has objects x, a Hom object $\mathcal A(x,y)$ in the base for every ordered pair, composition morphisms from $\mathcal A(y,z)\otimes\mathcal A(x,y)$ to $\mathcal A(x,z)$, and identity morphisms from I to $\mathcal A(x,x)$. Associativity and identity diagrams use the base's coherence maps. When the base is locally small, its ordinary maps from I into these Hom objects form the Hom sets of an underlying ordinary category.

Take the Boolean poset with tensor given by conjunction and unit true. A Hom object is a truth value expressing whether x relates to y. Identity requires reflexivity, and composition requires transitivity. Antisymmetry is not an enriched axiom, so this base describes preorders rather than only partially ordered sets. The base is symmetric, but that does not require its enriched relation to be symmetric: a directed order is already a counterexample.

Use the monoidal poset $([0,\infty],\ge,+,0)$, where an arrow from r to s exists when r is at least s. Composition becomes the usual triangle inequality, and the unit forces a zero self-distance. The allowed infinity satisfies the standard extended addition rule. Neither symmetry nor separation of distinct points at positive distance is required. These are Lawvere generalized metric spaces, whose axioms are weaker than those of ordinary symmetric, separated metric spaces.

For a finite directed graph with nonnegative edge weights, shortest-path distances provide such an enrichment whenever all distances are interpreted with infinity for missing routes. Concatenating two paths proves the triangle inequality after taking shortest lengths. Raw edge weights need not satisfy it, so taking their closure changes some entries. The associated ordinary category has a morphism exactly when the distance is zero, since a base map from the unit zero to a distance exists precisely in that case.