By Eugenia L. Cheng
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Additional info for Category Theory [Lecture notes]
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.
Category Theory [Lecture notes] by Eugenia L. Cheng