[Paper Review] The snail lemma for internal groupoids

Sandra Mantovani, Giuseppe Metere+1 more|arXiv (Cornell University)|Jul 4, 2017

Homotopy and Cohomology in Algebraic Topology13 references3 citations

TL;DR

This paper generalizes the Gabriel-Zisman and Brown exact sequences to internal groupoids in pointed regular categories with reflexive coequalizers, introducing a 'snail lemma' that constructs a six-term exact sequence using strong h-kernels, connected components ($\pi_0$), and automorphism functors ($\pi_1$). The key contribution is a unified framework where the sequence reduces to the classical snake lemma in abelian or semi-abelian categories.

ABSTRACT

We establish a generalized form both of the Gabriel-Zisman exact sequence associated with a pointed functor between pointed groupoids, and of the Brown exact sequence associated with a fibration of pointed groupoids. Our generalization consists in replacing pointed groupoids with groupoids internal to a pointed regular category with reflexive coequalizers.

Motivation & Objective

• To extend the Gabriel-Zisman and Brown exact sequences from pointed groupoids to groupoids internal in a pointed regular category with reflexive coequalizers.
• To unify the construction of exact sequences in non-abelian homological algebra by introducing a generalized 'snail lemma' for internal functors.
• To show that the strong h-kernel of an internal functor corresponds to the categorical kernel when the functor is an internal fibration, generalizing Brown's result.
• To establish that the functors $\pi_0$ and $\pi_1$ preserve exactness in the context of internal groupoids.

Proposed method

• Construct a six-term exact sequence in the base category $\mathcal{A}$ using the strong h-kernel of an internal functor, $\pi_0$, and $\pi_1$ functors.
• Utilize strong h-pullbacks and the universal property of strong h-kernels to define the exact sequence in the 2-category $\mathbf{Grpd}(\mathcal{A})$.
• Prove that when the internal functor is a fibration, the strong h-kernel coincides with the categorical kernel, leveraging results from a companion paper on fibrations of internal groupoids.
• Apply the pseudo-adjunction between the $[-]_1$ and $\pi_1$ functors to show that $\pi_1$ preserves exactness and that $g_1$ is an isomorphism.
• Use the regularity of $\mathcal{A}$ and the properness of the target groupoid to show that $\pi_0$ preserves exactness via regular epimorphisms.
• Demonstrate that the canonical comparison maps $g_0$ and $g_1$ are regular epimorphism and isomorphism respectively, ensuring exactness of the $\pi_0$ and $\pi_1$ sequences.

Experimental results

Research questions

• RQ1 Can the Gabriel-Zisman exact sequence be generalized beyond pointed groupoids to internal groupoids in a pointed regular category with reflexive coequalizers?
• RQ2 How does the strong h-kernel of an internal functor relate to the categorical kernel when the functor is an internal fibration?
• RQ3 Under what conditions do the functors $\pi_0$ and $\pi_1$ preserve exactness in the context of internal groupoids?
• RQ4 In what sense does the snail lemma unify the classical snake lemma and the Brown sequence in abelian or semi-abelian categories?

Key findings

• The snail lemma constructs a six-term exact sequence in the base category $\mathcal{A}$ from an internal functor, involving the strong h-kernel, $\pi_0$, and $\pi_1$.
• When the internal functor is an internal fibration, the strong h-kernel reduces to the categorical kernel, recovering Brown's exact sequence.
• $\pi_1$ preserves exactness: the comparison map $g_1$ is an isomorphism, ensuring exactness of the $\pi_1$-sequence.
• If the target groupoid is proper, $\pi_0$ preserves exactness: the comparison map $g_0$ is a regular epimorphism.
• The functors $\pi_0$ and $\pi_1$ preserve exactness of the snail sequence, as shown by the regular epimorphism and isomorphism properties of the comparison maps.
• In abelian or semi-abelian categories, the snail lemma reduces to the classical snake lemma, unifying non-abelian and abelian homological tools.