The mapping cone of a chain map

Depends on

• Chain map
• The shift of a chain complex
• The category of complexes in an additive category is additive

Used by

• Minimal complexes and the perverse truncations in the real type-A Soergel category Definition
• Signed totalization of graded Aₘ-bimodule actions Definition
• The bounded projective homotopy category Cₘ and the two shifts Definition
• The canonical inclusion and projection for a mapping cone Definition
• The cone triangle of a chain map Definition
• The mapping cylinder of a chain map Definition
• The relative homology of a chain map Definition
• The twist complexes Rᵢ and Rᵢ⁻¹ Definition
• Whitehead torsion of a finite CW homotopy equivalence Definition
• Euler class of a two-term mapping cone Example
• Point operations and the open closed constructible line Example
• The cone of multiplication by m on the integers Example
• The cone triangle of multiplication by m Example
• FALSE: the mapping-cone differential needs no minus sign False statement
• A lifted finite CW equivalence has a contractible group-ring mapping cone Lemma
• Acyclic directions of the Čech–Godement double complex Lemma
• An elementary CW expansion has zero Whitehead torsion Lemma
• Basis-change, direct-sum and based exact-sequence formulas Lemma
• Bounded two-sided projective bimodule complexes act on Cₘ Lemma
• Euler classes of Rouquier complexes are homotopy invariant and multiplicative Lemma
• Finite products and comparison cones have homological finite type Lemma
• Homological and internal shifts on K₀(Cₘ) Lemma
• Homotopic maps have chain-isomorphic mapping cones Lemma
• Koszul Append One Element Mapping Cone Identification Lemma
• The bounded projective comparison for the derived category Lemma
• The handle complex of an h-cobordism is contractible over the group ring, with an explicit contraction Lemma
• The Khovanov-Seidel grid resolutions of the vertex modules Lemma
• The mapping-cone differential squares to zero Lemma
• The three-cone calculation for a composite chain map Lemma
• An exact functor carries mapping-cone sequences to mapping-cone sequences Proposition
• Cones preserve chain-homotopy equivalences of arrows Proposition
• Isomorphic chain maps have isomorphic cones Proposition
• The cone construction commutes with shift up to the canonical sign isomorphism Proposition
• The cone of the zero map is the direct sum with a shift Proposition
• The quotient of the mapping cylinder by its source is the mapping cone Proposition
• A chain map is a homotopy equivalence exactly when its cone is contractible Theorem
• A chain map is a quasi-isomorphism exactly when its cone is acyclic Theorem
• A morphism of chain maps induces a chain map of cones Theorem
• Bounded bimodule tensor is associative, unital, and compatible with cones Theorem
• Composition and based-pair sum formulas for Whitehead torsion Theorem

…and 6 more results.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
• The Stacks Project, Section 13.9: Cones and termwise split sequences (standard reference, not scraped)