adjunction

Meaning

The act of joining; the thing joined or added.
The joining of personal property owned by one to that owned by another.
Given a pair of categories $\mathcal{C}$ and $\mathcal{D}$ : an anti-parallel pair of functors $F:\mathcal{C}\rightarrow \mathcal{D}$ and $G:\mathcal{D}\rightarrow \mathcal{C}$ and a natural transformation $\eta:\mbox{id}_C \rightarrow GF$ called “unit” such that for any object $A \in \mathcal{C}$ , for any object $B \in \mathcal{D}$ , and for any morphism $f:A\rightarrow GB$ , there is a unique morphism $g:FA\rightarrow B$ such that $Gg \circ \eta_A = f$ . (Note: there is another natural transformation called “counit” as well but its existence may be derived by theorem.) The pair of functors express a similarity between the pair of categories which is weaker than that of an equivalence of categories.

From Latin adjunctio, from adjungere: compare French adjonction, and see adjunct.

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