It is well-known that the category of presheaf functors is complete and cocomplete, and that the Yoneda embedding into the presheaf category preserves products. However, the Yoneda embedding does not preserve coproducts. It is perhaps less well-known that if we restrict the codomain of the Yoneda embedding to the full subcategory of limit-preserving functors, then this embedding preserves colimits, while still enjoying most of the other useful properties of the Yoneda embedding. We call this modified embedding the Lambek embedding. The category of limit-preserving functors is known to be a reflective subcategory of the category of all functors, i.e., there is a left adjoint for the inclusion functor. In the literature, the existence of this left adjoint is often proved non-constructively, e.g., by an application of Freyd's adjoint functor theorem. In this paper, we provide an alternative, more constructive proof of this fact. We first explain the Lambek embedding and why it preserves coproducts. Then we review some concepts from multi-sorted algebras and observe that there is a one-to-one correspondence between product-preserving presheaves and certain multi-sorted term algebras. We provide a construction that freely turns any presheaf functor into a product-preserving one, hence giving an explicit definition of the left adjoint functor of the inclusion. Finally, we sketch how to extend our method to prove that the subcategory of limit-preserving functors is also reflective.
翻译:众所周知, 预树叶配方的类别是完整和完整的, 而Yoneda 嵌入到预树叶类中保存产品。 然而, Yoneda 嵌入并不保存共产物。 也许不那么广为人知的是, 如果我们限制Yoneda 嵌入到限值配方的整个子类中, 那么这个嵌入会保存共同限值, 同时仍然享受Yoneda 嵌入的多数其他有用特性。 我们称之为这个修改过的嵌入 Lambek 嵌入的变异体。 限制保存的易燃剂类中, 已知的是所有配方类别中的一种反射子类, 即 Yoneda 嵌入的组合。 在文献中, 这个左侧连接的存在往往被证明为非交替性, 例如, 由Freyd 的粘合配方的配方的配方 。 在本文中, 我们提供了一个替代的, 更具建设性的证明这个事实。 我们首先解释 将 限制的调制制式配方嵌入和 将一个最终定义 保存一个预产值 。