By A. I. Kostrikin, I. R. Shafarevich
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Additional resources for Algebra I. Basic notions of algebra
Conversely, given such an interpretation of T in E, there is, up to natural isomorphism, a unique regular functor: C(T ) → E with the property that it maps the standard interpretation of T in C(T ) to the given one in E. Exercise 84 This exercise constructs the “free regular category on a given category C”. Given C, which is not assumed to be regular (or to have finite limits), let L be the language of C as before: it has a sort C for every object C of C, and a function symbol (f : C → D) for every arrow of C.
In the diagram C D G K / DT C b_bb bbbb F GT |||||> bbbb || bb |||| G bbb ~|||| F T | D we have that KF = F T and GT K = G. Moreover, the functor K is unique with this property. This leads us to define a category T -Adj of adjunctions C o that GF = T . A map of such T -adjunctions from C o F G F G / D to C o / D such F G / D is a functor K : C → C satisfying KF = F and G K = G. What we have proved about T -Alg can be summarized by saying that the adjunction DT o FT GT / D is a terminal object in T -Adj.
In this chapter we assume that we have such a choice, as well as a distinguished terminal object 1; and we assume 62 also that for each object X we have a specified right adjoint to the functor (−) × X, which we write as (−)X (Many authors write X ⇒ (−), but I think that overloads the arrows notation too much). Objects of the form Z X are called exponents. We have the unit ηY,X Y → (Y × X)X and counit εY,X YX ×X → Y of the adjunction (−) × X (−)X .
Algebra I. Basic notions of algebra by A. I. Kostrikin, I. R. Shafarevich