Monoidal adjunction

Suppose that and are two monoidal categories. A monoidal adjunction between two lax monoidal functors

and

is an adjunction between the underlying functors, such that the natural transformations

and

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that

is a lax monoidal functor such that the underlying functor has a right adjoint . This adjunction lifts to a monoidal adjunction if and only if the lax monoidal functor is strong.

gollark: It's called 5G because it's fifth generation because it comes after 4G.
gollark: No.
gollark: I don't like it. We use a BT router with that "feature" at home and I cannot figure out how to turn it off and it *annoys me slightly*.
gollark: Self-driving cars should probably not be using the mobile/cell network just for communicating with nearby cars, since it adds extra latency and complexity over some direct P2P thing, and they can't really do things which rely on constant high-bandwidth networking to the internet generally, since they need to be able to not crash if they go into a tunnel or network dead zone or something.
gollark: My problem isn't *that* (5G apparently has improvements for more normal frequencies anyway), but that higher bandwidth and lower latency just... isn't that useful and worth the large amount of money for most phone users.

See also

  • Every monoidal adjunction defines a monoidal monad .
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