[Paper Review] A characterization of final functors between internal groupoids in exact categories

Alan S. Cigoli|arXiv (Cornell University)|Nov 29, 2017

Homotopy and Cohomology in Algebraic Topology6 references3 citations

TL;DR

This paper characterizes final functors between internal groupoids in Barr-exact categories by proving that such a functor is final if and only if it is full and essentially surjective. The characterization extends the classical result from ordinary groupoids to the internal setting using categorical tools like the décalage functor and pullback stability in exact categories.

ABSTRACT

This paper provides several characterizations of final functors between internal groupoids in Barr-exact categories. In particular, it is proved that an internal functor between groupoids is final if and only if it is full and essentially surjective.

Motivation & Objective

• To extend the classical characterization of final functors in ordinary groupoids to the internal setting within Barr-exact categories.
• To provide an internal counterpart to the well-known result that a functor between groupoids is final if and only if it is full and essentially surjective.
• To establish a precise categorical characterization of final functors using the décalage functor and pullback stability in exact categories.
• To bridge a gap in the literature by formally proving that fullness and essential surjectivity characterize finality in internal groupoids.

Proposed method

• Utilizes the décalage functor to construct the comprehensive factorization of internal functors in Barr-exact categories.
• Applies pullback stability of discrete fibrations and opfibrations in exact categories to analyze internal functor properties.
• Employs the comma category construction internally via pullbacks to define finality in the internal context.
• Uses the morphism π₀(F) and the arrow ψ_F in a commutative diagram to characterize finality via regular epimorphisms and isomorphisms.
• Relies on Corollary 2.3 and Proposition 2.4 to relate fullness and essential surjectivity to the isomorphism and regular epimorphism properties of π₀(F) and ψ_F.
• Establishes equivalence between finality and the conjunction of π₀(F) being an isomorphism and ψ_F being a regular epimorphism.

Experimental results

Research questions

• RQ1 What is the internal characterization of final functors between internal groupoids in a Barr-exact category?
• RQ2 Does the classical characterization—final if and only if full and essentially surjective—hold in the internal setting?
• RQ3 How do the properties of fullness and essential surjectivity relate to the structural properties of the décalage functor and pullbacks?
• RQ4 Can the comprehensive factorization system for internal functors be characterized via final functors and discrete fibrations in exact categories?
• RQ5 What conditions on π₀(F) and ψ_F ensure that an internal functor is final?

Key findings

• An internal functor between groupoids in a Barr-exact category is final if and only if it is full and essentially surjective.
• The morphism π₀(F) is an isomorphism precisely when the functor is essentially surjective, as established via Proposition 2.4.
• The arrow ψ_F is a regular epimorphism if and only if the functor is full, as shown through the equivalence with φ_F in the diagram.
• Finality is equivalent to π₀(F) being an isomorphism and ψ_F being a regular epimorphism, as formalized in Corollary 4.3.
• The décalage functor enables the construction of the comprehensive factorization and provides a key tool for analyzing finality in the internal context.
• Pullback stability of discrete fibrations and the use of regular epimorphisms ensure that the characterization holds in the exact categorical setting.