nLab opposite adjunction




Given a pair of adjoint functors

𝒟RL𝒞 \mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} \mathcal{C}

there is an induced adjunction of opposite functors between their opposite categories of the form

𝒞 opL opR op𝒟 op. \mathcal{C}^{op} \underoverset {\underset{L^{op}}{\longrightarrow}} {\overset{R^{op}}{\longleftarrow}} {\;\;\;\;\bot\;\;\;\;} \mathcal{D}^{op} \,.

Hence where LL was the left adjoint, its opposite becomes the right adjoint, and dually for RR.

This is immediate from the definition of opposite categories and the characterization of adjoint functors via the corresponding hom-isomorphism.

The adjunction unit of the opposite adjunction has as components the components of the original adjunction counit, regarded in the opposite category, and dually:

ϵ d R opL op:R opL op(d)(η d RL) opd,AAAAAAη c L opR op:c(ϵ c LR) opL opR op(c). \epsilon^{R^{op} L^{op}}_{d} \;\colon\; R^{op}\circ L^{op}(d) \xrightarrow{\;\; \big( \eta^{R L}_d \big)^{op} \;\;} d \,, {\phantom{AAAAAA}} \eta^{L^{op} R^{op}}_{c} \;\colon\; c \xrightarrow{\;\; \big( \epsilon^{L R}_c \big)^{op} \;\;} L^{op} \circ R^{op}(c) \,.

Last revised on February 19, 2023 at 13:50:57. See the history of this page for a list of all contributions to it.