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By Peter Johnstone, D. Mehrle

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Finally, we should check that given any cone pβ j : pA, αq Ñ Dpjq | j P ob Jq over D in CT , this factors through the limit cone corresponding to L. But we know that there is a unique β : A Ñ L in C such that λ j β “ β j for all j. Once again, Tβ TA α TL λ β A L So pλ j : pL, λq Ñ Dpjq | j P ob Jq is a limit cone in CT . (ii) Simialrly, if D : J Ñ CT and pλ j : GDpjq Ñ L | j P ob Jq is a colimit for GD in C, then pTλ j : TGDpjq Ñ TL | j P ob Jq is also a colimit because T preserves colimits, so the composites TGDpjq δj λj GDpjq ÝÑ L induce a unique λ : TL Ñ L.

A cone over it consists of C and the arrows as in the diagram below h C A k B D Equivalently, this is a way of completing this diagram to a commutative square h C f k g B A D A limit for the diagram is called a pullback of the pair f , g. (iv) Colimits of shape Jop are called pushouts; they can similarly be constructed from coproducts and coequalizers. 8. (i) Products / coproducts in Set are cartesian products / disjoint unions. Likewise in Top. In algebraic categories like Gp, Ab, Ring, R-Mod, etc.

This is called the Eilenberg-Moore category. 5. The forgetful functor CT Ñ C has left adjoint FT : C Ñ CT and the adjunction induces the monad T. f Proof. We define FT A “ pTA, µ A q and FT pA Ý Ñ Bq “ T f , which is an algebra by Equation 11 and Equation 13. Clearly GT FT “ T, and we have a natural transformation η : 1C Ñ GT FT . We define ε : FT GT Ñ 1CT by ε p A,αq “ α : TA Ñ A. This is a homomorphism by Equation 15, and natural by Equation 16. We just need to check the triangular identities now.

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Category Theory by Peter Johnstone, D. Mehrle

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