By Max Kelly
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53) A(ST A, A ); this follows from Yoneda on setting A = A and composing both legs with j, since (V n)( A ) = 1. Because n is an isomorphism, we conclude that T is fully faithful if and only if is an isomorphism. In consequence, a right adjoint T is fully faithful exactly when T0 is so. /G B is called an equivalence, and we As in any 2-category, a V-functor T : A G / A and isomorphisms η : 1 ∼ write T : A B, if there is some S : B = T S and ∼ ∼ ρ : ST = 1. Replacing ρ (still in any 2-category) by the isomorphism : ST = 1 given by /G B.
15) whenever {F, G}, T {F, G}, and T G exist. This conclusion may be expressed as: partial right adjoints in so far as they are defined, preserve limits. /G C, the conclusion In the special case where S in fact has a right adjoint T : B becomes: right adjoints preserve all limits that exist. 3 Limits in functor categories; double limits and iterated limits /G [A, B] correspond We now consider limits in a functor category [A, B]. Let G : K /G B. ; so that, in particular, /G B. Let F : K /G V. (G−)A denotes P (−, A) : K Then, if {F, (G−)A} exists for each A, the limit {F, G} in [A, B] exists, and we have {F, G}A = {F, (G−)A}.
24) between the set of (projective) cones over G with vertex B and the set of cones over B(B, G−) with vertex the one-point set 1. The ordinary, or classical, or conical limit /G B is the representing object lim G [resp. colim G] in [resp. 25) existing when the right side here is representable. 5) that lim G = {∆1, G}, colim G = ∆1 G. 26) Thus the classical limit of G : K limit of G indexed by ∆1 : K the limit-cone /G B, often called a limit indexed by K, is in fact the /G Set. 2) now corresponds to µK : lim G /G GK.