Abstract
AbstractThis paper explores versions of the Yoneda Lemma in settings founded upon FM sets. In particular, we explore the lemma for three base categories: the category of nominal sets and equivariant functions; the category of nominal sets and all finitely supported functions, introduced in this paper; and the category of FM sets and finitely supported functions. We make this exploration in ordinary, enriched and internal settings. We also show that the finite support of Yoneda natural transformations is a theorem for free.
Highlights
The purpose of this paper is to establish some instances of the Yoneda Lemma in settings involving both nominal sets and FM sets
In working with our future applications, we found that we needed to perform various concrete computations, which were only enabled by unravelling the abstract details of the enriched category theory
Work This paper grew from plans to use categorical gluing to establish the conservativity of pure NLC over Nominal Equational Logic (NEL)
Summary
The purpose of this paper is to establish some instances of the Yoneda Lemma in settings involving both nominal sets and FM sets. Let us recall some of the basic ingredients of Nominal Equational Logic (NEL), with the notation below based on Figure 5 of Clouston and Pitts (2007). Crole has shown (Crole 1993, 1996) that the existence of ∼=1 can be reduced to showing that the functor category V D is a ccc and that D(A, B) ∼=2 V D (YA, YB) Isomorphisms such as ∼=2, and the cartesian closure of functor categories like V D , can be established using instances of the (enriched) Yoneda Lemma.
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