Theorems · Definition · category theory

HomotopicalAlgebra.FibrantObject.HoCat.iResolutionObj

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : HomotopicalAlgebra.ModelCategory C] → (X : C) → X ⟶ HomotopicalAlgebra.FibrantObject.HoCat.resolutionObj X

This is a trivial cofibration X ⟶ resolutionObj X where resolutionObj X is a choice of a fibrant resolution of X.

Defined in: Mathlib.AlgebraicTopology.ModelCategory.FibrantObjectHomotopy
Cited by: 7 results in Mathlib
Foundations: Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category HomotopicalAlgebra.ModelCategory

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Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites4

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement · cited by 32,603
HomotopicalAlgebra.ModelCategorystatement and proof · cited by 141
HomotopicalAlgebra.FibrantObject.HoCat.resolutionObjstatement · cited by 7

Cited by9

Results whose statement or proof uses this declaration.

HomotopicalAlgebra.FibrantObject.HoCat.resolutionMap_facstatement · cited by 2
HomotopicalAlgebra.bijective_rightHomotopyClassToHomproof · cited by 2
HomotopicalAlgebra.FibrantObject.HoCat.ιCompResolutionNatTransproof · cited by 1
HomotopicalAlgebra.FibrantObject.HoCat.exists_resolution_mapstatement and proof · cited by 1
HomotopicalAlgebra.FibrantObject.HoCat.resolutionMap_fac_assocstatement and proof · cited by 0
HomotopicalAlgebra.FibrantObject.HoCat.resolutionObj_hom_extstatement and proof · cited by 0
HomotopicalAlgebra.FibrantObject.HoCat.weakEquivalence_resolutionMap_iffproof · cited by 0
HomotopicalAlgebra.FibrantObject.HoCat.ιCompResolutionNatTrans_appstatement · cited by 0
HomotopicalAlgebra.FibrantObject.HoCat.resolutionCompToLocalizationNatTransproof · cited by 0