1
Introduction
▶
1.1
Main results
1.2
Structure of the paper
1.3
A formalisation
2
Preliminaries
▶
2.1
Discretisation
2.2
Bizero objects and null morphisms
2.12
2-monomorphisms and 2-epimorphisms
2.17
2-kernels and 2-cokernels
3
2-kernels, 2-cokernels and normality
▶
3.1
2-z-exactness and normality
3.5
Composition and cancellation
3.10
Short 2-exact sequences
3.21
The normal image factorisation
3.24
Morphisms of short 2-exact sequences
3.27
The Normal Short Five Lemma
4
Exact sequences and the Pure Snake Lemma
▶
4.1
The canonical comparison
4.3
Exactness
4.6
Dinversion
4.12
Normal chain complexes and homology
4.16
Homological self-duality
4.20
The Third Isomorphism Property
4.23
Exactness via homology
4.26
The Pure Snake Lemma
5
Bipullbacks and the squares of a morphism of short 2-exact sequences
▶
5.1
Bipullbacks
5.6
The two squares
6
The Snake Lemma in 2-di-exact 2-categories
▶
6.1
2-di-exact 2-categories
6.8
The statement, and the construction
6.11
The connecting 1-cell
6.16
Exactness at the outer positions
6.19
The general case
6.23
The classical Snake Lemma
7
Naturality
▶
7.1
Pure configurations and their morphisms
7.5
2-naturality of the comparison
7.8
2-naturality of the snake sequence
8
Two-dimensional models
▶
8.2
Serre subcategories are the 2-kernels
8.7
The saturation, and the failure of
(DI2)
8.14
Saturated abelian categories
8.18
A criterion on subobjects
8.23
Modules and coherent sheaves
8.30
A locally ordered model: modular lattices
9
The Snake Lemma without self-duality
▶
9.1
The two hypotheses
9.7
Why \(\textup{(DPN)}\) alone is not enough
9.9
Two normal morphisms
9.15
The connecting 1-cell
9.18
The theorem
9.21
\(\mathit{AbCat}\) is homologically self-dual but does not satisfy \(\textup{(DPN)}\)
9.28
A two-dimensional model: Hilbert lattices
10
Bibliography
Dependency graph
A two-categorical Snake Lemma — blueprint
E. Caviglia, L. Mesiti and T. Van der Linden
1
Introduction
1.1
Main results
1.2
Structure of the paper
1.3
A formalisation
2
Preliminaries
2.1
Discretisation
2.2
Bizero objects and null morphisms
2.12
2-monomorphisms and 2-epimorphisms
2.17
2-kernels and 2-cokernels
3
2-kernels, 2-cokernels and normality
3.1
2-z-exactness and normality
3.5
Composition and cancellation
3.10
Short 2-exact sequences
3.21
The normal image factorisation
3.24
Morphisms of short 2-exact sequences
3.27
The Normal Short Five Lemma
4
Exact sequences and the Pure Snake Lemma
4.1
The canonical comparison
4.3
Exactness
4.6
Dinversion
4.12
Normal chain complexes and homology
4.16
Homological self-duality
4.20
The Third Isomorphism Property
4.23
Exactness via homology
4.26
The Pure Snake Lemma
5
Bipullbacks and the squares of a morphism of short 2-exact sequences
5.1
Bipullbacks
5.6
The two squares
6
The Snake Lemma in 2-di-exact 2-categories
6.1
2-di-exact 2-categories
6.8
The statement, and the construction
6.11
The connecting 1-cell
6.16
Exactness at the outer positions
6.19
The general case
6.23
The classical Snake Lemma
7
Naturality
7.1
Pure configurations and their morphisms
7.5
2-naturality of the comparison
7.8
2-naturality of the snake sequence
8
Two-dimensional models
8.2
Serre subcategories are the 2-kernels
8.7
The saturation, and the failure of
(DI2)
8.14
Saturated abelian categories
8.18
A criterion on subobjects
8.23
Modules and coherent sheaves
8.30
A locally ordered model: modular lattices
9
The Snake Lemma without self-duality
9.1
The two hypotheses
9.7
Why \(\textup{(DPN)}\) alone is not enough
9.9
Two normal morphisms
9.15
The connecting 1-cell
9.18
The theorem
9.21
\(\mathit{AbCat}\) is homologically self-dual but does not satisfy \(\textup{(DPN)}\)
9.28
A two-dimensional model: Hilbert lattices
10
Bibliography