Title: Left adjoint to precomposition in elementary doctrines Show Chinese title

Title: 预复合的左伴随在初等理论中

Abstract: It is well-known in universal algebra that adding structure and equational axioms generates forgetful functors between varieties, and such functors all have left adjoints. The category of elementary doctrines provides a natural framework for studying algebraic theories, since each algebraic theory can be described by some syntactic doctrine and its models are morphism from the syntactic doctrine into the doctrine of subsets. In this context, adding structure and axioms to a theory can be described by a morphism between the two corresponding syntactic doctrines, and the forgetful functor arises as precomposition with this last morphism. In this work, given any morphism of elementary doctrines, we prove the existence of a left adjoint of the functor induced by precomposition in the doctrine of subobjects of a Grothendieck topos. Show Chinese abstract

Abstract: 在普遍代数中，众所周知，添加结构和等式公理会在种类之间生成遗忘函子，这样的函子都有左伴随。 初等教义的范畴为研究代数理论提供了自然的框架，因为每个代数理论都可以通过某种语法教义来描述，其模型是从语法教义到子集教义的态射。 在此背景下，向一个理论添加结构和公理可以通过两个相应语法教义之间的态射来描述，而遗忘函子则作为对该最后态射的前复合出现。 在本工作中，给定任何初等教义的态射，我们证明了在格罗滕迪克拓扑的子对象教义中，由前复合诱导的函子存在左伴随。