Categories for the Working Mathematician by Saunders Mac Lane

Author:Saunders Mac Lane
Language: eng
Format: epub, pdf
Publisher: Springer-Verlag Wien 2012
Published: 2014-06-02T16:00:00+00:00

2. Algebras for a Monad

The natural question, “Can every monad be defined by a suitable pair of adjoint functors?” has a positive answer, in fact there are two positive answers provided by two suitable pairs of adjoint functors. The first answer (due to Eilenberg-Moore [1965]) constructs from a monad 〈T, η, μ〉 in X a category of XT of “T-algebras” and an adjunction X ⇀ XT which defines 〈T, η, μ〉 in X. Formally, the definition of a T-algebra is that of a set on which the “monoid” T acts (cf. the introduction).

Definition. If T = 〈T, η, µ〉 is a monad in X, a T-algebra 〈x, h〉 is a pair consisting of an object x ∈ X (the underlying object of the algebra) and an arrow h : T x→x of X (called the structure map of the algebra) which makes both the diagrams (1)

commute. (The first diagram is the associative law, the second the unit law.) A morphism f : 〈x, h〉→〈x′, h′〉 of T-algebras is an arrow f : x→x′ of X which renders commutative the diagram (2)

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