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?


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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