Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 5

Let $(\mathcal{V},\otimes ,I)$ be a symmetric monoidal category. A $\mathcal{V}$-enriched category has objects, hom-objects $\mathcal{C}(A,B)\in \mathcal{V}$, composition morphismsand unit morphisms $I\to \mathcal{C}(A,A)$ satisfying the associative and unit diagrams. Its underlying ordinary category has hom-sets

If $\mathcal{V}$ is closed, take its internal hom $[A,B]$ as hom-object. Composition is the transpose of evaluationand the unit is the transpose of $I\otimes A\cong A$. This is the self-enrichment of a closed symmetric monoidal category.

For posets $A,B$, let $B^A$ be the poset of monotone maps $A\to B$ ordered pointwise. Evaluationis monotone. A monotone map $h:C\times A\to B$ curries to the monotone mapand this correspondence is natural and invertible. Thus the cartesian closed category of posets has exponentials $B^A$.

Identities are self-adjoint. If $f⊣g$ and $f^{\prime }⊣g^{\prime }$, thenwith the second inequality written after inserting the two units in the appropriate order. Hence $f^{\prime }f⊣g{g}^{\prime }$, so left adjoints form a subcategory.

In $\mathbf{Poset}$, these are exactly monotone maps possessing right adjoints, equivalently lower adjoints; when all joins exist, they are precisely the arbitrary-join-preserving maps.

In the inclusion-ordered category of relations, a relation $R:A⇝B$ is left adjoint exactly when it is total and single-valued. It is therefore the graph of a function, and its right adjoint is $R^{\mathrm{op}}$. This is the left adjoint relation is a function criterion.