Residuals are “exponential objects for a monoidal category”: they are compatible with the monoidal product . Because is (in general) not symmetric, we distinguish a left residual and a right residual. In a symmetric monoidal category, these are the same.
Recall the definition of an exponential object for a cartesian category:
Let be two objects of the cartesian category. An exponential object is an object with data such that for every object , there is a one-to-one mapping between morphisms
and morphisms
The backwards direction is given by , namely . Going back and forth is also required to be the identity: for , we must have .
Let be two objects of a monoidal category.
A left residual of by is an object together with an evaluation map and for objects a transformation from morphisms
to morphisms
such that , and for every morphism .
A right residual of by is an object together with an evaluation map and for objects a transformation from morphisms
to morphisms
such that , and for every morphism .
Mnemonic for the notation: looks like dividing by on the right, and looks like dividing by on the left.
Left and right residuals are unique up to isomosphism.
Last revised on February 15, 2017 at 08:26:59. See the history of this page for a list of all contributions to it.