# Algebra Seminar

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Given a commutative square with the top arrow surjective, the Snake Lemma produces a six term exact sequence, constructed from the kernel and cokernel of the morphisms involved. Its version for pointed groupoids, also called the Brown sequence, constructs a six term exact sequence from a fibration, its kernel and their homotopy invariants. We generalize this result for a *-fibration internal to a regular pointed category. Through the normalization process, we get back the classical Snake Lemma.

Due to the assumption of surjectivity, the Snake Lemma is somehow asymmetric. To overtake this asymmetry, Vitale established the so called Snail Lemma, which associates a six term exact sequence from any commutative square. In the case where the top arrow is a surjection, we get back the Snake Lemma. Its denormalised version, also called the Gabriel-Zisman sequence, constructs from any functor F between internal groupoids an exact sequence using the strong homotopy kernel of F. With a careful analysis of *-fibrations and the comparison between kernels and strong homotopy kernels, one can deduces the Brown sequence from the Gabriel-Zisman sequence.

This is a joint work with Sandra Mantovani, Guiseppe Metere and Enrico Vitale.