symmetric monoidal (∞,1)-category of spectra
For a category and a - bimodule , an algebra for is given by a functor and an extranatural transformation , where is constant at the point. is called the carrier of the algebra. A morphism of -algebras is given by a natural transformation such that .
If is the one-object category, an algebra is given by an object in and an element . A morphism between two algebras and is then a morphism in such that , these both being elements of .
There is an an obvious forgetful functor into from the category of algebras for , which sends each algebra to its carrier and each algebra morphism to its underlying morphism in ; among other properties, this functor is always faithful and conservative.
In fact, the category , together with its forgetful functor , has the universal property of an Eilenberg-Moore object, namely that of being the universal -algebra. Specifically, it is a terminal object in the category whose objects are functors equipped with an extranatural transformation . For such an extranatural transformation consists of, for every , an element , such that for every morphism in , we have . This is precisely the data of a functor lying over .
One version of Yoneda's lemma says that for a profunctor there is a bijection between extranatural transformations and natural transformations . So there are bijections
Algebras and coalgebras for endofunctors are special cases of algebras for bimodules; specifically, an algebra for an endofunctor is an algebra for the bimodule , while a coalgebra for is an algebra for the bimodule .
A natural transformation between functors and from to is a section of the forgetful functor into from the category of algebras for the bimodule . That is, it gives every object of the structure of an algebra for in such a way as that every morphism of has the property of being an algebra morphism between the algebras on its domain and codomain.
A natural numbers object (in the weak, unparametrized sense) in a category with terminal object is an initial object in the category of algebras for the bimodule . If has binary coproducts, then this is of course the same as an initial algebra for the endofunctor .