Definition Simple Optics [lcc-0004]

Let \mathcal {C} be a monoidal category. Then \mathsf {Optic}(\mathcal {C}) is a category where

• Objects are pairs \binom {A}{X} of objects in \mathcal {C}
• The set of morphisms is \operatorname {\mathrm {Hom}}\left (\binom {A}{X},\binom {B}{Y}\right ) = \int ^{M \in \mathcal {C}} \mathcal {C}(X, M \otimes Y) \times \mathcal {C}(M \otimes B, A)

Note that many sources, including Categories of optics, write the forwards direction on top, rather than on the bottom as we have done here. Writing the forwards direction on the bottom is in line with the comparison with dependent optics, where the backwards pass is indexed in some sense over the forwards pass.