![]() Note that although we are speaking of internal models of a syntactic 0-theory in an arbitrary semantic 0-theory (in the same 0-doctrine = 1-theory), these semantic 0-theories are not themselves arbitrary internal models of the 1-theory in question, but specifically models in the “canonical” 1-category Set Set. interpretations of each generator x ∈ G x\in G as an element of M M such that the relations in R R hold in M M. Each individual monoid (like ( ℕ, +, 0 ) (\mathbb \to M, for any monoid M M, are equivalent to “internal models” of ( G, R ) (G,R) in M M, i.e. This is a 1-theory, as you might guess from the fact that monoids form a 1-category. Let’s start with a simple example of a theory: the theory of monoids.
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