14 Simplicial Methods

Section 14.1: Introduction
Section 14.2: The category of finite ordered sets
- Definition 14.2.1
- Lemma 14.2.2
- Lemma 14.2.3
- Lemma 14.2.4
Section 14.3: Simplicial objects
- Definition 14.3.1
- Lemma 14.3.2
- Remark 14.3.3
- Example 14.3.4
- Example 14.3.5
- Lemma 14.3.6
Section 14.4: Simplicial objects as presheaves
- Equation 14.4.0.1
Section 14.5: Cosimplicial objects
- Definition 14.5.1
- Lemma 14.5.2
- Remark 14.5.3
- Example 14.5.4
- Example 14.5.5
- Example 14.5.6
- Lemma 14.5.7
Section 14.6: Products of simplicial objects
- Definition 14.6.1
- Lemma 14.6.2
Section 14.7: Fibre products of simplicial objects
- Definition 14.7.1
- Lemma 14.7.2
Section 14.8: Pushouts of simplicial objects
- Definition 14.8.1
- Lemma 14.8.2
Section 14.9: Products of cosimplicial objects
- Definition 14.9.1
- Lemma 14.9.2
Section 14.10: Fibre products of cosimplicial objects
- Definition 14.10.1
- Lemma 14.10.2
Section 14.11: Simplicial sets
- Definition 14.11.1
- Example 14.11.2
- Lemma 14.11.3
- Example 14.11.4
- Lemma 14.11.5
Section 14.12: Truncated simplicial objects and skeleton functors
- Definition 14.12.1
Section 14.13: Products with simplicial sets
- Equation 14.13.0.1
- Definition 14.13.1
- Lemma 14.13.2
- Lemma 14.13.3
- Lemma 14.13.4
Section 14.14: Hom from simplicial sets into cosimplicial objects
- Definition 14.14.1
Section 14.15: Hom from cosimplicial sets into simplicial objects
- Definition 14.15.1
- Lemma 14.15.2
Section 14.16: Internal Hom
Section 14.17: Hom from simplicial sets into simplicial objects
- Definition 14.17.1
- Lemma 14.17.2
- Lemma 14.17.3
- Lemma 14.17.4
- Lemma 14.17.5
Section 14.18: Splitting simplicial objects
- Definition 14.18.1
  - Equation 14.18.1.1
- Lemma 14.18.2
- Lemma 14.18.3
- Lemma 14.18.4
- Lemma 14.18.5
- Lemma 14.18.6
- Lemma 14.18.7 slogan
- Lemma 14.18.8
- Lemma 14.18.9
Section 14.19: Coskeleton functors
- Equation 14.19.0.1
- Example 14.19.1
- Lemma 14.19.2
- Lemma 14.19.3
- Lemma 14.19.4
- Lemma 14.19.5
- Lemma 14.19.6
- Remark 14.19.7
- Remark 14.19.8
- Remark 14.19.9
- Lemma 14.19.10
- Remark 14.19.11
- Lemma 14.19.12
- Lemma 14.19.13
- Lemma 14.19.14
- Lemma 14.19.15
Section 14.20: Augmentations
- Definition 14.20.1
- Lemma 14.20.2
- Lemma 14.20.3
- Remark 14.20.4
Section 14.21: Left adjoints to the skeleton functors
- Lemma 14.21.1
- Lemma 14.21.2
- Lemma 14.21.3
- Lemma 14.21.4
- Lemma 14.21.5
- Remark 14.21.6
- Lemma 14.21.7
- Lemma 14.21.8
- Lemma 14.21.9
- Lemma 14.21.10
- Lemma 14.21.11
Section 14.22: Simplicial objects in abelian categories
- Lemma 14.22.1
- Lemma 14.22.2
- Definition 14.22.3
- Lemma 14.22.4
- Lemma 14.22.5
Section 14.23: Simplicial objects and chain complexes
- Lemma 14.23.1
- Lemma 14.23.2
- Lemma 14.23.3
- Lemma 14.23.4
- Lemma 14.23.5
- Lemma 14.23.6
- Remark 14.23.7
- Lemma 14.23.8
- Lemma 14.23.9
Section 14.24: Dold-Kan
- Lemma 14.24.1
- Lemma 14.24.2
- Theorem 14.24.3
Section 14.25: Dold-Kan for cosimplicial objects
- Lemma 14.25.1
Section 14.26: Homotopies
- Definition 14.26.1
- Lemma 14.26.2
- Example 14.26.3
- Remark 14.26.4
- Remark 14.26.5
- Definition 14.26.6
- Example 14.26.7
- Lemma 14.26.8
- Lemma 14.26.9
- Lemma 14.26.10
Section 14.27: Homotopies in abelian categories
- Equation 14.27.0.1
- Lemma 14.27.1
- Lemma 14.27.2
Section 14.28: Homotopies and cosimplicial objects
- Definition 14.28.1
  - Equation 14.28.1.1
- Remark 14.28.2
- Lemma 14.28.3
- Lemma 14.28.4 slogan
- Lemma 14.28.5
- Lemma 14.28.6 slogan
- Lemma 14.28.7
Section 14.29: More homotopies in abelian categories
- Lemma 14.29.1
- Lemma 14.29.2
- Lemma 14.29.3
Section 14.30: Trivial Kan fibrations
- Definition 14.30.1
- Lemma 14.30.2
- Lemma 14.30.3
- Lemma 14.30.4
- Lemma 14.30.5
- Lemma 14.30.6 slogan
- Lemma 14.30.7
- Lemma 14.30.8
Section 14.31: Kan fibrations
- Definition 14.31.1
- Lemma 14.31.2
- Lemma 14.31.3
- Lemma 14.31.4
- Lemma 14.31.5
- Lemma 14.31.6
- Lemma 14.31.7
- Lemma 14.31.8
- Lemma 14.31.9
Section 14.32: A homotopy equivalence
- Lemma 14.32.1
- Lemma 14.32.2
- Lemma 14.32.3
Section 14.33: Preparation for standard resolutions
- Example 14.33.1
- Lemma 14.33.2
- Example 14.33.3
- Lemma 14.33.4
- Lemma 14.33.5
- Lemma 14.33.6
Section 14.34: Standard resolutions
- Situation 14.34.1
  - Equation 14.34.1.1
- Lemma 14.34.2
- Lemma 14.34.3
- Example 14.34.4
- Example 14.34.5
- Example 14.34.6
- Example 14.34.7
- Example 14.34.8