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Subtypes and bounded quantification from a fibred perspective

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Abstract

A general categorical description of subtyping σ < σ′ and of bounded quantification ∀α<: σ.τ and ∃α <: σ.τ is presented in terms of fibrations. In fact, we shall generalize these bounded quantifiers to “constrained quantifiers” ∀α[σ <: σ′].τ and ∃α[σ <: σ′].τ. In these cases one quantifies over those type variables α for which σ(α) <: σ′(α) holds. Semantically we distinguish three levels: types τ, which are fibered over (depend on) subtypings σ <: σ′, which in turn are fibred over (depend on) kinds K. In this setting we can describe constrained quantification ∀α[σ <: σ′]. (−) and ∃α[σ <: σ′]. (−) as right and left adjoints to the weakening functor which adds the (dummy) hypothesis σ <: σ′ to an appropriate context. This shows that, like ordinary quantifiers, these constrained (and hence especially bounded) quantifiers are adjoints.

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