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By Eugenia L. Cheng

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Iff (X ↓ G) has initial object GAFT where we define P : (X ↓ G) D to be the obvious forgetful functor. So:   P : (X ↓ G) D creates small limits.  (X ↓ G) be a diagram. We need to show that, if PD has a limit cone, then Let D : ‫މ‬ there is a cone cI (V DI)I∈‫މ‬ PcI in (X ↓ G) such that (PV PDI)I∈‫ މ‬is a limit for PD in D, and that any such cone is itself a limit for D in (X ↓ G).  Suppose PD : ‫މ‬ D has a limit cone, say (L cI PDI)I∈‫ މ‬: L cI PDI  G preserves small limits, so (GL GcI cI PDI cI PDI GPDI)I∈‫ މ‬is a limit for GPD in C.

Now, we have:X FX ηX GFX 1FX GFf FX X GFX Ff FX f 1FX ηX X FX GFX But we have transposed twice, and hence we have equality as required. Similarly for ε.  Given F G, we call η : 1C GF the unit and ε : FG 1D the counit of the adjunction.  Free  U : Gp forgetful. For example: Set has a left adjoint F U, where F(S) gives the free group on S; so we have Gp(FS, G) ∼ = Set(S, U(G)) Vect which forgets the multiplicative structure; we have F  U : Alg the free algebra on V. }  U : Ab Gp has a left adjoint “free abelianization”: GAB = G/[G, G].

And hence K is full and faithful by the previous result.  If D has and G preserves coequalisers for all G-split coequaliser pairs, then K is essentially surjective.  θ Given any algebra TA A, we seek Y ∈ D such that KY ∼ = TA T3 A TµA Tθ T2 A T2 θ µTA µA TA Tθ A in CT . ). , FGFA K ◦ UT = G). FA Fθ FA is a G-split coequaliser pair, and K of it is the pair in () (since So it has a coequaliser in D, FGFA FA Fθ FA h Y () say. We show that K of this coequaliser is a coequaliser of the same parallel pair we started with.

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Category Theory [Lecture notes] by Eugenia L. Cheng


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