Pseudomonoid

A monoid is a set equipped with an associative, unital, binary operation. A monoid object in a monoidal category is an object equipped with morphisms which one thinks of as the unit and binary operation. A monoid object in $\mathsf{Set}$ is precisely the first notion of monoid.

A monoidal category? is a categorification of the notion of monoid. A pseudomonoid is the corresponding concept for monoid objects. Thus, you can consider pseudomonoid objects in various monoidal 2-categories. A pseudomonoid in $\mathsf{Cat}$ is a monoidal category.

It is worth noting that a monoid object in $\mathsf{Cat}$ is a very strict sort of monoidal category. This is not very practical, as it does not appear “in nature” very often. Pseudomonoids can then be seen as precisely the weakening of monoid objects one needs to capture the examples found in nature.

The definition follows that of monoidal category nearly exactly, of course except that it does not have to be a category equipped with functors.

- Let $X$ be a category. A pseudomonoid in $\Fib(X)$, the 2-category of Grothendieck fibrations over X, is precisely a (fibre-wise) monoidal fibration?.

- Brian Day? and Ross Street?, Monoidal bicategories and Hopf algebroids,
*Adv. Math.*, 129(1):99–157, 1997.

category: mathematical methods