Abstract
We define here a notion of internal copy and of weak internal copy of a category. We will then determine some families of categories having an internal copy or a weak internal copy. We will consider categories of definable classes of first-order theories and we will see that the notion of internal copy is related to the notion of numerical existence property.
Highlights
A category C can host internal algebraic structures such as monoids, groups, rings, etc
An internal category is defined as an internal graph having a composition arrow and an identity arrow making some diagrams commute
If p : Cop → InfSL is a doctrine, that is a contravariant functor from C to the category of inf-semilattices, we can consider the internal categories in the base category Qp of its elementary quotient completion
Summary
A category C can host internal algebraic structures such as monoids, groups, rings, etc. Notational conventions In this paper we adopt the following convention: whenever we say that C is a finitely complete category, we mean that C is endowed with a distinct terminal object 1 and with an explicit choice of pullbacks (Pb(f, g), π0f,g, π1f,g) for every pair of arrows f and g having the same codomain In this case, if k0 and k1 are arrows in C such that f ◦ k0 = g ◦ k1, we denote by k0, k1 f,g the only arrow from ∂0(k0) = ∂0(k1) to Pb(f, g) such that πif,g ◦ k0, k1 f,g = ki for i = 0, 1. We will omit the subscripts and superscripts when they will be clear from the context
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