The snail lemma and the long homology sequence

This paper establishes a homotopical version of the snail lemma and applies it to sequentiable families of arrows to derive a long exact sequence in a general category, which generalizes the classical long homology sequence when the category is abelian.

The Gist

The Big Picture: Fixing a Broken Chain

Imagine you are a detective trying to solve a mystery involving a long chain of events. In mathematics, this "chain" is called a chain complex. It's a sequence of objects (like boxes) connected by arrows (like doors) where the output of one door becomes the input of the next.

Usually, mathematicians want to know: "What gets stuck in the middle?" If a door leads to a room, but nothing comes out the other side, that "stuck" stuff is called homology. It's the residue, the leftover evidence.

The classic problem in this field is: If you have a short, perfect chain of events, how do you figure out the "stuck stuff" for the whole long chain?

For decades, the standard tool to solve this was the Snake Lemma. Think of the Snake Lemma as a very specific, rigid wrench. It only works if your chain is a "short exact sequence"—a very neat, perfectly aligned trio of boxes. If your chain is messy or doesn't fit that perfect trio shape, the Snake Lemma can't help you.

The New Tool: The Snail Lemma

This paper introduces a new, more flexible tool called the Snail Lemma.

• The Snake vs. The Snail: The Snake is rigid and straight. The Snail is flexible; it can curl around obstacles. The authors show that you don't need a perfect "short exact sequence" to find the homology. You can start with any morphism (any connection) between chains, and the Snail Lemma will still work.
• The Analogy: Imagine the Snake Lemma is a straight ruler. It can only measure things that are perfectly straight. The Snail Lemma is a flexible measuring tape. It can measure a straight line, but it can also curl around a curve, a knot, or a weird shape and still give you an accurate measurement.

The Secret Ingredient: "Sequentiable Families"

To make the Snail Lemma work, the authors had to invent a new concept called a Sequentiable Family of Arrows.

• The Problem: In the old way of doing things, mathematicians looked at the "boxes" (the objects) in the chain. But the Snail Lemma works better if you look at the "doors" (the arrows) and how they connect.
• The Solution: A "Sequentiable Family" is like a conveyor belt system. Instead of just looking at the boxes sitting on the belt, you are looking at the entire mechanism: the belt, the rollers, and the specific way the items move from one station to the next.
• Why it helps: By focusing on the movement (the arrows) rather than just the objects, the authors created a new mathematical playground (a category called Seq(A)). In this playground, the rules of "homotopy" (which is a fancy word for "flexible deformation") are much friendlier. It's like moving from a rigid concrete floor to a trampoline; you can bounce and twist in ways you couldn't before.

The Magic Trick: Unrolling the Sequence

Here is the clever part of the paper's method:

1. The Six-Term Sequence: When you apply the Snail Lemma in this new "conveyor belt" world, you get a compact, six-step sequence. It's like a short, neat puzzle piece.
2. Unrolling: Because the authors built this system so carefully, they can take that single six-step puzzle piece and "unroll" it.
3. The Result: When you unroll it, it stretches out into a Long Exact Sequence. This is the famous, long list of equations that connects all the "stuck stuff" (homology) of the original chain.

The Metaphor: Imagine you have a coiled garden hose (the six-term sequence). It's compact and easy to hold. The authors found a way to uncoil it perfectly so that it stretches out into a long, straight line (the long homology sequence) without any kinks or leaks.

Why Does This Matter?

1. It's More General: The old Snake Lemma was like a key that only opened one specific type of lock. The Snail Lemma is a master key that opens almost any lock in this mathematical universe.
2. It Unifies Things: The paper shows that the "long homology sequence" (a famous result in algebra) isn't just a special case for perfect chains. It's actually a natural consequence of a more general rule that applies to messy, imperfect chains too.
3. It's a Bridge: The authors built a bridge between two different worlds:
   • Chain Complexes: The traditional way of doing algebra.
   • Internal Groupoids: A more abstract, geometric way of thinking about math (related to shapes and paths).
     By showing that the Snail Lemma works in the "conveyor belt" world, they proved that these two different ways of thinking are actually talking about the same underlying truth.

Summary in One Sentence

The authors invented a flexible new tool (the Snail Lemma) and a new way of looking at connections (Sequentiable Families) to show that you can calculate the "leftover evidence" of a long mathematical chain, even if the chain is messy and doesn't follow the strict rules required by the old methods.

Technical Summary

Based on the provided paper "The snail lemma and the long homology sequence" by Julia Ramos González and Enrico M. Vitale, here is a detailed technical summary.

1. Problem Statement

In classical homological algebra within an abelian category, the construction of a long exact homology sequence typically relies on the Snake Lemma. However, the Snake Lemma requires the starting point to be a short exact sequence of chain complexes.

The authors address two main limitations:

1. Generality: They seek to construct long exact sequences starting from an arbitrary morphism of chain complexes, not just short exact sequences.
2. Structural Unification: They aim to unify the construction of long exact sequences with higher-dimensional exact sequences found in homotopy theory and the study of groupoids/crossed modules, which are often governed by the Snail Lemma (a generalization of the Snake Lemma).
3. Homotopical Structure: The standard category of chain complexes ($Ch(A)$) possesses a homotopy structure that is "too weak" to directly apply the homotopical version of the Snail Lemma because it fails the reduced interchange condition required for strong homotopy kernels and cokernels.

2. Methodology

The authors employ a categorical approach involving categories with nullhomotopies (introduced by Grandis) and the introduction of a new category called $Seq(A)$.

A. Theoretical Framework

• Nullhomotopy Structures: They work within a pointed category $\mathcal{B}$ equipped with a structure of nullhomotopies $\Theta$. This structure allows for the definition of homotopy kernels ($N(g)$) and homotopy cokernels ($C(g)$).
• The Snail Lemma: They utilize the homotopical Snail Lemma, which constructs a six-term exact sequence for any morphism $g$ in $\mathcal{B}$, provided $\mathcal{B}$ has strong homotopy kernels/cokernels and satisfies the reduced interchange condition.
• Exactness Classes: They define a class of "surjection-like" arrows $S$ (satisfying specific closure and cancellation properties) to define $S$-exactness.

B. The Novel Construction: $Seq(A)$

To overcome the limitations of $Ch(A)$, the authors introduce the category $Seq(A)$ of sequentiable families of arrows.

• Definition: A sequentiable family $h_\bullet$ is a sequence of pairs of arrows $(h_n, i_n)$ where $i_n$ connects the cokernel of $h_{n+1}$ to the kernel of $h_n$.
• Nullhomotopies in $Seq(A)$: They equip $Seq(A)$ with a nullhomotopy structure $\Theta_\Delta$ (analogous to the arrow category $Arr(A)$). Crucially, this structure satisfies the reduced interchange condition, unlike the standard structure on $Ch(A)$.
• The Functor $F$: They construct a functor $F: Ch(A) \to Seq(A)$ that maps a chain complex to a sequentiable family. This functor effectively "unrolls" the complex into a form where the homotopy structure is robust.

C. The Strategy

1. Apply Snail Lemma in $Seq(A)$: Apply the homotopy Snail Lemma to a morphism in $Seq(A)$. This yields a six-term exact sequence within the category $Seq(A)$.
2. Unrolling: "Unroll" this six-term sequence level-by-level to obtain a long exact sequence in the base category $A$.
3. Comparison: Compare this new long sequence with the classical long homology sequence derived from short exact sequences of complexes.

3. Key Contributions

1. Homotopical Snail Lemma for General Morphisms: The paper establishes that the Snail Lemma can be used to generate long exact sequences from any morphism of chain complexes, removing the requirement for the morphism to be part of a short exact sequence.
2. Introduction of $Seq(A)$: The definition of the category of sequentiable families provides a new categorical environment where the homotopy structure is "convenient" (satisfies reduced interchange), allowing for the application of the Snail Lemma where it previously failed in $Ch(A)$.
3. Recovery of Homology: They demonstrate that for a complex $C_\bullet$, the homology objects $H_n(C_\bullet)$ can be recovered as the cokernels (or kernels) of specific arrows within the associated sequentiable family $F(C_\bullet)$.
4. Unification of Concepts: The work bridges the gap between the classical Snake Lemma (abelian case, short exact sequences) and the Snail Lemma (homotopy theory, arbitrary morphisms), showing they are special cases of a broader homotopical framework.

4. Key Results

• The Six-Term Sequence in $Seq(A)$: For any morphism $f_\bullet: h_\bullet \to h'_\bullet$ in $Seq(A)$, there exists a six-term exact sequence involving homotopy kernels and cokernels:
  $$N(0_{N(f)}) \to N(0_h) \to N(0_{h'}) \to \pi_0(N(f)) \to \pi_0(h) \to \pi_0(h')$$
  This sequence is $S$-exact under specific conditions (e.g., if the category is regular and protomodular).

• The Long Exact Sequence in $A$: By unrolling the six-term sequence in $Seq(A)$, the authors derive a long exact sequence in the base category $A$:
  $$\dots \to H_{n+1}(B) \to H_{n+1}(C) \to \text{Coker}(h^P_{n+1}) \to H_n(B) \to H_n(C) \to \dots$$
  Here, $h^P$ represents the kernel component of the morphism in $Seq(A)$.

• Equivalence with Classical Results:

  • If the morphism in $Seq(A)$ arises from an extension of chain complexes (a short exact sequence), the resulting long exact sequence coincides exactly with the classical long homology sequence.
  • If the base category $A$ is abelian, the assumptions on the objects being "proper" (a condition related to regular epimorphisms) are automatically satisfied, making the result universally applicable to abelian categories.

• Functoriality of Homotopies: The functor $F: Ch(A) \to Seq(A)$ extends to a morphism of categories with nullhomotopies, preserving the homotopical structure. This implies that quasi-isomorphisms in $Ch(A)$ correspond to weak equivalences in $Seq(A)$.

5. Significance

• Generalization: The paper significantly generalizes the scope of homological algebra. It proves that the machinery of long exact sequences does not strictly depend on short exact sequences but can be derived from arbitrary morphisms via the Snail Lemma.
• Structural Insight: It provides a deeper structural understanding of why the Snake Lemma works and how it relates to higher-dimensional homotopy theory. By shifting the focus from homology objects to homology arrows (within $Seq(A)$), the authors reveal a more natural setting for homotopical algebra.
• Applicability: The results hold in regular protomodular categories (which include abelian categories, groups, and Lie algebras), not just abelian categories. This makes the findings relevant for non-abelian homological algebra and the study of internal groupoids.
• Methodological Innovation: The introduction of $Seq(A)$ serves as a powerful tool for translating problems in chain complexes into a setting where strong homotopy limits and colimits behave well, potentially opening new avenues for research in homotopy theory and categorical algebra.

In summary, the paper successfully reconstructs the long homology sequence using the Snail Lemma within a newly defined category of sequentiable families, thereby unifying classical homological algebra with modern homotopical methods and extending these results to non-abelian contexts.