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It is often written that "(the syntactic category of) a Lawvere theory" but I don't know the difference.

To begin with, I don't understand the concept "syntactic category" very well. Sometimes I read the definition but can't figure out, and I try to begin at understanding syntactic category of Lawvere theory.

Could you tell me what is syntactic category in this case?

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There are different kinds of syntactic category, corresponding to different kinds of logical theory. For instance:

  • Algebraic (a.k.a. equational) theories → Categories with finite products
  • Cartesian (a.k.a. essentially algebraic or finite limit) theories → Categories with finite limits
  • Coherent (a.k.a. existential positive) theories → Coherent categories
  • Intuitionistic first-order theories → Heyting categories
  • Classical first-order theories → Boolean categories

The basic idea, however, is the same. For a given theory to be interpreted in categories of a certain type, the associated syntactic category is a category of that type that contains a universal model of that theory. For example, given an algebraic theory $\mathbb{T}$, the syntactic category of $\mathbb{T}$ is a category $\mathcal{C}[\mathbb{T}]$ with finite products such that, for every category $\mathcal{S}$ with finite products, there is a natural equivalence between finite-product-preserving functors $\mathcal{C}[\mathbb{T}] \to \mathcal{S}$ and models of $\mathbb{T}$ in $\mathcal{S}$. In other words, the Lawvere theory of $\mathbb{T}$ is the syntactic category of $\mathbb{T}$.

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