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  • C and D are two ∞-categories (quasi-categories),
  • $F : C \to D$ and $G : C \to D$ are two functors (i.e. 0-simplices in the ∞-category of functors Fun(C,D), which is just the simplicial mapping complex),
  • $a : F \to G$ and $b : F \to G$ are two natural transformations (i.e. 1-simplices in Fun(C,D)),
  • at each object (0-simplex) x of C, $a_x : F(x) \to G(x)$ and $b_x : F(x) \to G(x)$ are equivalent.

Does it follow that a and b are equivalent in Fun(C,D)?

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up vote 6 down vote accepted

No, not necessarily, since a transformation involves "higher structure" in addition to its 1-cell components. For example, let C be the "walking arrow" category with two objects 0 and 1 and one nonidentity morphism from 0 to 1, and let D be an abelian group regarded as a (2,1)-category with one object and one morphism, and thereby as an (∞,1)-category. There is exactly one functor from C to D. An endo-natural-transformation of this functor consists of giving, for each object of C, a morphism in D, and for each morphism in C, a 2-cell in D, subject to certain axioms.

There is only one morphism in D, so any two such transformations will have equal 1-cell components, but their 2-cells might be different. The axioms say that the 2-cell corresponding to an identity morphism is the identity, and that the 2-cell corresponding to a composite is the composite of the 2-cells corresponding to the individual factors; thus for the C and D above, to give a transformation is exactly to give an arbitrary element of the abelian group D (the 2-cell component corresponding to the single nonidentity arrow of C).

Now two transformations are equivalent when there is a modification between them, which consists of for each object of C, a 2-cell in D between the corresponding 1-cell components, and for each morphism of C, a 3-cell relating the 2-cell components etc. Since 3-cells in D are all identities, this latter means that there is a diagram that must commute. Since C has one object, a modification thus consists of giving a single element of the abelian group D, and the diagram which must commute means that it conjugates the element corresponding to the first transformation to the element corresponding to the second one. Thus, the equivalence classes of such transformations (all of which have the same 1-cell component) are the conjugacy classes of D.

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