By van Oosten J.
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Extra resources for Basic category theory
T X → Y is the composite Now f˜ T (ηX ) T (f ) h T X → T 2X → T Y → Y Since f is a T -algebra map, this is T (ηX ) µX f T (X) → T 2 (X) → T (X) → Y which is f by the monad laws. Conversely, g¯ ˜ : X → Y is the composite ηX T (g) h X → TX → TY → Y By naturality of η and the fact that (Y, h) is a T -algebra, we conclude that g¯ ˜ = g. So we have a natural 1-1 correspondence C(X, U T (Y, h)) T -Alg(F T (X), (Y, h)) and our adjunction. Note that the composite U T F T is the functor T , and that the unit η of the adjunction is the unit of T ; the proof that for the counit ε of F T U T we have that U T εF T T 2 = UT F T UT F T → UT F T = T is the original multiplication µ, is left to you.
Many categories, as we have seen, have a forgetful functor to Set which has a left adjoint. So the forgetful functor preserves limits, and since these can easily be computed in Set, one already knows the “underlying set” of the vertex of the limiting cone one wants to compute. 51 Does a converse to the theorem hold? e. given G : C → D which preserves all limits; does G have a left adjoint? In general no, unless C is sufficiently complete, and a rather technical condition, the “solution set condition” holds.
In this situation, let Q ⊆ P be the image of T , with the ordering inherited from P . We have maps r : P → Q and i : Q → P such that ri is the identity on Q and ir = T : P → P . For x ∈ P , y ∈ Q we have x ≤ i(y) ⇔ r(x) ≤ y (check); so r i and the operation T arises from this adjunction. 1 Algebras for a monad Given a monad (T, η, µ) on a category C, we define the category T -Alg of T algebras as follows: • Objects are pairs (X, h) where X is an object of C and h : T (X) → X is an arrow in C such that T 2 (X) T (h) / T (X) µX T (X) h /X h ηX / T (X) Xi ii ii and ii ii h idX " X commute; f • Morphisms: (X, h) → (Y, k) are morphisms X → Y in C for which T (X) T (f ) / T (Y ) h k X f /Y commutes.