Editing Adjoint functors (section)

== Terminology and notation ==

The terms ''[[wikt:adjoint|adjoint]]'' and ''[[wikt:adjunct|adjunct]]'' are both used, and are [[cognate]]s: one is taken directly from Latin, the other from Latin via French.  In the classic text ''Categories for the Working Mathematician'', [[Saunders Mac Lane|Mac Lane]] makes a distinction between the two. Given a family

: <math>\varphi_{cd}: \mathrm{hom}_{\mathcal{C}}(Fd,c) \cong \mathrm{hom}_{\mathcal{D}}(d,Gc)</math>

of hom-set bijections, we call <math>\varphi</math> an '''adjunction''' or an '''adjunction between <math> F </math> and <math> G </math>'''. If <math>f</math> is an arrow in <math> \mathrm{hom}_{\mathcal{C}}(Fd,c) </math>, <math>\varphi f</math> is the right '''adjunct''' of <math>f</math> (p.&nbsp;81). The functor <math> F </math> is '''left adjoint''' to <math>G</math>, and <math>G</math> is '''right adjoint''' to <math>F</math>. (Note that <math>G</math> may have itself a right adjoint that is quite different from <math>F</math>; see below for an example.)

In general, the phrases "<math> F </math> is a left adjoint" and "<math> F </math> has a right adjoint" are equivalent. We call <math>F</math> a left adjoint because it is applied to the left argument of <math>\mathrm{hom}_{\mathcal{C}}</math>, and <math>G</math> a right adjoint because it is applied to the right argument of <math>\mathrm{hom}_{\mathcal{D}}</math>.

If ''F'' is left adjoint to ''G'', we also write
:<math>F\dashv G.</math>

The terminology comes from the [[Hilbert space]] idea of [[adjoint operator]]s <math>T</math>, <math>U</math> with <math>\langle Ty,x\rangle = \langle y,Ux\rangle</math>, which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.<ref>{{cite arXiv|arxiv=q-alg/9609018|first=John C.|last=Baez|title=Higher-Dimensional Algebra II: 2-Hilbert Spaces|year=1996}}</ref>