Simplicial and dendroidal homotopy theory:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cham, Switzerland
Springer
[2022]
|
| Series: | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics
75 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xx, 612 Seiten Diagramme 24 cm |
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MARC
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| 245 | 1 | 0 | |a Simplicial and dendroidal homotopy theory |c Gijs Heuts ; Ieke Moerdijk |
| 264 | 1 | |a Cham, Switzerland |b Springer |c [2022] | |
| 300 | |a xx, 612 Seiten |b Diagramme |c 24 cm | ||
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| 490 | 1 | |a Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |v 75 | |
| 650 | 4 | |a Homotopy theory | |
| 700 | 1 | |a Moerdijk, Ieke |d 1958- |0 (DE-588)1067170103 |4 aut | |
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Record in the Search Index
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| adam_text | Contents
Part I The Elementary Theory of Simplicial and Dendroidal Sets
Operads 0 0 00 ccc cece cece nee e et eneneeeenes 3
EL Operads 0 c cece cece ete tetanenes 3
1 2 Algebras for Operads vee bebe cence ete eees 10
121 Definitions and Examples 2 0-0 22c eee eeee 10
122 Free Algebras 0 00 0 0 ccc ccc center n ees 14
123 Change of Operad 200 000 e eee cee eee 15
1 3 Trees ccc cence teen e erent ete enneee 16
1 4 Alternative Definitions for Operads 000 00 cee eee 20
[5 Free Operads 200 0c cece cence nee nett nnees 23
1 6 The Tensor Product of Operads 000 e eee eee ee eee 26
1 7 The Boardman-Vogt Resolution of an Operad 29
1 8 Configuration Spaces and the Fulton-MacPherson Operad 37
1 9 Configuration Spaces and the Operad of Little Cubes 44
Simplicial Sets 0 o cece cee ete eee eens 49
2 1 The Simplex Category Ao 00 0 0 49
2 2 Simplicial Sets and Geometric Realization 065 53
2 3 The Geometric Realization as a Cell Complex - 57
2 4 Simplicial Sets as a Category of Presheaves 222 62
2 5 Products of Simplicial Sets and ShuflleMaps 70
2 6 Simplicial Spaces and Bisimplicial Sets - --2rerr2000 77
261 Simplicial Spaces - 2eese esse rennen een nenn 77
262 Bisimplicial Sets 0 cece ene e eee eee ees 78
2 7 Simplicial Categories and Simplicial Operads -- 80
271 Internal Versus Enriched Categories and Operads 81
272 Simplicial Categories 06c erence eee eee eee 82
273 Boardman—Vogt Resolution 0 00: e cece e eee eee 83
274 Homotopy-Coherent Nerve 0006 0: c eee eee eee 85
275 Simplicial Operads -- 6 cece cece ene eee eens 86
xviii
Contents
276 The Barratt-Eccles Operad 62-0 ee eee eee, 87
277 The Simplicial Boardman—Vogt Resolution of an Operad g9
Dendroidal Sets 2 020 006 cece cee eee een 91
3 1 Trees 0 eee eens Beene eee eee, 9]
3 2 The Category © of Trees 20 eee een 95
3 3 Faces and Degeneracies in 2 © 2 06 2 eee eee ee eee eee 98
331 Outer Faces eee ee ene neces eee eee eeeaye 99
332 Inner Faces 2 2 02 cee ee ete e eee 99
333 Degeneracies ----eecessenersneneen nennen 100
3 34 Codendroidal Identities ------sacssenaennene nn 100
335 Factorization of Morphisms Between Trees 101
336 Some Limits and Colimits in Q 000, 104
34 Dendroidal Sets 0 00000 cc et eee ee eens 106
3 5 Categories Related to Dendroidal Sets - -- -- 0 02, 112
351 Dendroidal Sets and Operads -- 00 2 113
352 Dendroidal Sets and Simplicial Sets 2 -2 114
353 Dendroidal Sets and Simplicial Operads nennen 115
354 Open Dendroidal Sets 250-2020 eee eee ee 115
355 Closed Dendroidal Sets 0 - 00002 cece eee eee 116
356 Uncoloured Dendroidal Sets - 0- 00 0 0 117
357 Dendroidal Sets and T-Sets 2 222202 ennereenn 118
3 6 Normal Dendroidal Sets and Skeletal Filtration - 119
3 7 Normal Monomorphisms and Normalization 5 124
Tensor Products of Dendroidal Sets -2 -02 000002- 133
4 1 Elementary Properties and Shuffles of Trees - -- 133
4 2 The Tensor Product of a Simplicial and a Dendroidal Set 144
4 3 Tensor Products and Normal Monomorphisms --+++ 146
4 4 Unbiased Tensor Products 0 00 cee eee eee teres 155
Kan Conditions for Simplicial Sets bce eee e eee ees 161
5 1 Kan Complexes and co-Categories 0200200000 eee 161
5 2 Fibrations Between Simplicial Sets 00 0 000 020 e eee 168
5 3 Saturated Classes and Anodyne Morphisms - -2--250+5 173
5 4 Products, Joins, and Spines of Simplices 2 : 0: 179
5 5 Fibrations Between Mapping Spaces 2 2 2-0: 186
5 6 Equivalences in oo-Categories 0 00 0 0000 c cece eet 191
5 7 Minimal o0-Categories and Minimal Kan Complexes -+-- 199
5 8 Minimal Fibrations Between @-Categories e ee ren 206
Kan Conditions for Dendroidal Sets 2 22002803* 21
6 1 Dendroidal Kan Complexes and ©-Operads --:+*° a
6 2 Fibrations and Anodyne Morphisms Between Dendroidal Sets --- 220
6 3 Tensor Products and Anodyne Morphisms - 227
6 4 Fibrations Between Mapping Spaces of Dendroidal Sets -----
Contents
6 5 Spines and Leaves of Trees
6 6 Joins of Trees
6 7 Equivatences in o0-Operads 2 2 2222 22 nn 252
6 8 Minimal Fibrations Between co-Operads
wee cee cee cece eee eee 255
Part If The Homotopy Theory of Simplicial and Dendroidal Sets
7 Model Categories 2 22 02 265
7 1 Axioms for a Model Category 0002---00---22 2 266
7 2 Some Background on Topological Spaces 2 -- 270
7 3, A Model Structure for Topological Spaces 00 -00 2715
7 4 Homotopies Between Morphisms in a Model Category 280
7 5 The Homotopy Category of a Model Category 287
7 6 Brown’s Lemma and Proper Model Categories 290
7 7 Transfer of Model Structures 2 2 2222 293
7 8 Homotopy Pushouts and the Cube Lemma - 297
8 Model Structures on the Category of Simplicial Sets 303
8 1 The Categorical Model Structure on Simplicial Sets 304
8 2 The Kan-Quillen Model Structure on Simplicial Sets 313
8 3 Quillen Adjunctions and Derived Functors een 318
84 Homotopy Groups of Simplicial Sets 222222022220200 330
8 5 Geometric Realizations and Fibrations - 22222220 337
8 6 The Equivalence Between Simplicial Sets and Topological Spaces 341
8 7 Categorical Weak Equivalences Between co-Categories 343
8 8 The Covariant Model Structure 600 00 0 00022 c eee 347
9 Three Model Structures on the Category of Dendroidal Sets 353
9 1 The A-Model Structure for Dendroidal Sets ----- -- 354
9 2 The Operadic Model Structure 2-0 20222 e eee 22 371
9 3 Open and Uncoloured Dendroidal Sets 222020200 2 378
9 4 The Relative A-Model Structure - -2 -=-2onneeeennnnnn 383
9 5 The Covariant Model Structure on Dendroidal Sets 387
9 6 The Absolute Covariant Model Structure ------ 2 -- 396
9 7 The Picard Model Structure -----err2sssoenneneennnnn 404
Part Ill The Homotopy Theory of Simplicial and Dendroidal Spaces
10 Reedy Categories and Diagrams of Spaces ---- - - 423
10 1 Reedy Categories 606 00 c cece eee e eect tent n ee ees 423
10 2 Reedy Fibrations 0 6 00 06 2c eee e eee cece eee eee eens 426
10 3 The Reedy Model Structure -- 20-00 s cence een eee eee 431
10 4 Simplicial Objects and Geometric Realization 435
10 5 Homotopy Colimits 20022 e cece eee eee eee eee ee eee 442
10 6 A Version of Quillen’s Theorem B - -----+--+-0+ eee eee 447
Contents
11 Mapping Spaces and Bousfield Localizations 453
11 1 Mapping Spaces 222 rennen 453
11 2 Common Models for Mapping Spaces 2 460
11 3 Left Bousfield Localizations 000 0 cceeee cesses 467
11 4 Existence of Left Bousfield Localizations 2 469
11 5 Localizable Sets of Morphisms - 000 000 ec eee eee, 474
12 Dendroidal Spaces and oo-Operads 222 nn 481
12 1 Dendroidal Segal Spaces uccecc ee 482
12 2 Complete Dendroidal Segat Spaces 2 2222 49]
12 3 Complete Weak Equivalences 2cen ce 498
12 4 The Tensor Product of Dendroidal Spaces 0 505
12 5 Closed Dendroidal Spaces 22022 509
12 6 Reduced Dendroidal Spaces 2 22222en een 514
12 7 Simplicial Spaces 22unueneee nenn 519
13 Left Fibrations and the Covariant Model Structure 523
13 1 The Covariant Model Structure on Dendroidal Spaces 524
13 2 Simplicial Systems of Model Categories 22222222222222 530
13 3 Homotopy Invariance of the Covariant Model Structure 540
13 4 The Homotopy Theory of Algebras 22 2222222222 544
13 5 Algebras and Left Fibrations 22 2 22 00 549
14 Simplicial Operads and o0-Operads 00 000 000 0 ee eee 55S
14 1 Simplicial Categories with Fixed Objecis 22 22 2222 356
14 2 Equivalences in Simplicial Categories 22 2222222022222 559
14 3 A Model Structure for Simplicial Operads 22 22cm 564
14 4 The Sparse Model Structure 2 222222222 eneeeeennnnenn 567
14 5 Simplicial Operads and Dendroidal Spaces 573
14 6 The Homotopy-Coherent Nerve 00020 2c cece cece eee ees 576
14 7 Operads with a Single Colour 2c2cnceneesnenaneneenn 581
14 8 Algebras for co-Operads and for Simplicial Operads 587
Epilogue 2 0 cc cee eee cece cn nees 59}
References 000 000 cece eee cece cee e pny enes 601
|
| adam_txt |
Contents
Part I The Elementary Theory of Simplicial and Dendroidal Sets
Operads 0 0 00 ccc cece cece nee e et eneneeeenes 3
EL Operads 0 c cece cece ete tetanenes 3
1 2 Algebras for Operads vee bebe cence ete eees 10
121 Definitions and Examples 2 0-0 22c eee eeee 10
122 Free Algebras 0 00 0 0 ccc ccc center n ees 14
123 Change of Operad 200 000 e eee cee eee 15
1 3 Trees ccc cence teen e erent ete enneee 16
1 4 Alternative Definitions for Operads 000 00 cee eee 20
[5 Free Operads 200 0c cece cence nee nett nnees 23
1 6 The Tensor Product of Operads 000 e eee eee ee eee 26
1 7 The Boardman-Vogt Resolution of an Operad 29
1 8 Configuration Spaces and the Fulton-MacPherson Operad 37
1 9 Configuration Spaces and the Operad of Little Cubes 44
Simplicial Sets 0 o cece cee ete eee eens 49
2 1 The Simplex Category Ao 00 0 0 49
2 2 Simplicial Sets and Geometric Realization 065 53
2 3 The Geometric Realization as a Cell Complex - 57
2 4 Simplicial Sets as a Category of Presheaves 222 62
2 5 Products of Simplicial Sets and ShuflleMaps 70
2 6 Simplicial Spaces and Bisimplicial Sets - --2rerr2000 77
261 Simplicial Spaces - 2eese esse rennen een nenn 77
262 Bisimplicial Sets 0 cece ene e eee eee ees 78
2 7 Simplicial Categories and Simplicial Operads -- 80
271 Internal Versus Enriched Categories and Operads 81
272 Simplicial Categories 06c erence eee eee eee 82
273 Boardman—Vogt Resolution 0 00: e cece e eee eee 83
274 Homotopy-Coherent Nerve 0006 0: c eee eee eee 85
275 Simplicial Operads -- 6 cece cece ene eee eens 86
xviii
Contents
276 The Barratt-Eccles Operad 62-0 ee eee eee, 87
277 The Simplicial Boardman—Vogt Resolution of an Operad g9
Dendroidal Sets 2 020 006 cece cee eee een 91
3 1 Trees 0 eee eens Beene eee eee, 9]
3 2 The Category © of Trees 20 eee een 95
3 3 Faces and Degeneracies in 2 © 2 06 2 eee eee ee eee eee 98
331 Outer Faces eee ee ene neces eee eee eeeaye 99
332 Inner Faces 2 2 02 cee ee ete e eee 99
333 Degeneracies ----eecessenersneneen nennen 100
3 34 Codendroidal Identities ------sacssenaennene nn 100
335 Factorization of Morphisms Between Trees 101
336 Some Limits and Colimits in Q 000, 104
34 Dendroidal Sets 0 00000 cc et eee ee eens 106
3 5 Categories Related to Dendroidal Sets - -- -- 0 02, 112
351 Dendroidal Sets and Operads -- 00 2 113
352 Dendroidal Sets and Simplicial Sets 2 -2 114
353 Dendroidal Sets and Simplicial Operads nennen 115
354 Open Dendroidal Sets 250-2020 eee eee ee 115
355 Closed Dendroidal Sets 0 - 00002 cece eee eee 116
356 Uncoloured Dendroidal Sets - 0- 00 0 0 117
357 Dendroidal Sets and T-Sets 2 222202 ennereenn 118
3 6 Normal Dendroidal Sets and Skeletal Filtration - 119
3 7 Normal Monomorphisms and Normalization 5 124
Tensor Products of Dendroidal Sets -2 -02 000002- 133
4 1 Elementary Properties and Shuffles of Trees - -- 133
4 2 The Tensor Product of a Simplicial and a Dendroidal Set 144
4 3 Tensor Products and Normal Monomorphisms --+++ 146
4 4 Unbiased Tensor Products 0 00 cee eee eee teres 155
Kan Conditions for Simplicial Sets bce eee e eee ees 161
5 1 Kan Complexes and co-Categories 0200200000 eee 161
5 2 Fibrations Between Simplicial Sets 00 0 000 020 e eee 168
5 3 Saturated Classes and Anodyne Morphisms - -2--250+5 173
5 4 Products, Joins, and Spines of Simplices 2 : 0: 179
5 5 Fibrations Between Mapping Spaces 2 2 2-0: 186
5 6 Equivalences in oo-Categories 0 00 0 0000 c cece eet 191
5 7 Minimal o0-Categories and Minimal Kan Complexes -+-- 199
5 8 Minimal Fibrations Between @-Categories e ee ren 206
Kan Conditions for Dendroidal Sets 2 22002803* 21
6 1 Dendroidal Kan Complexes and ©-Operads --:+*° a
6 2 Fibrations and Anodyne Morphisms Between Dendroidal Sets --- 220
6 3 Tensor Products and Anodyne Morphisms - 227
6 4 Fibrations Between Mapping Spaces of Dendroidal Sets -----
Contents
6 5 Spines and Leaves of Trees
6 6 Joins of Trees
6 7 Equivatences in o0-Operads 2 2 2222 22 nn 252
6 8 Minimal Fibrations Between co-Operads
wee cee cee cece eee eee 255
Part If The Homotopy Theory of Simplicial and Dendroidal Sets
7 Model Categories 2 22 02 265
7 1 Axioms for a Model Category 0002---00---22 2 266
7 2 Some Background on Topological Spaces 2 -- 270
7 3, A Model Structure for Topological Spaces 00 -00 2715
7 4 Homotopies Between Morphisms in a Model Category 280
7 5 The Homotopy Category of a Model Category 287
7 6 Brown’s Lemma and Proper Model Categories 290
7 7 Transfer of Model Structures 2 2 2222 293
7 8 Homotopy Pushouts and the Cube Lemma - 297
8 Model Structures on the Category of Simplicial Sets 303
8 1 The Categorical Model Structure on Simplicial Sets 304
8 2 The Kan-Quillen Model Structure on Simplicial Sets 313
8 3 Quillen Adjunctions and Derived Functors een 318
84 Homotopy Groups of Simplicial Sets 222222022220200 330
8 5 Geometric Realizations and Fibrations - 22222220 337
8 6 The Equivalence Between Simplicial Sets and Topological Spaces 341
8 7 Categorical Weak Equivalences Between co-Categories 343
8 8 The Covariant Model Structure 600 00 0 00022 c eee 347
9 Three Model Structures on the Category of Dendroidal Sets 353
9 1 The A-Model Structure for Dendroidal Sets ----- -- 354
9 2 The Operadic Model Structure 2-0 20222 e eee 22 371
9 3 Open and Uncoloured Dendroidal Sets 222020200 2 378
9 4 The Relative A-Model Structure - -2 -=-2onneeeennnnnn 383
9 5 The Covariant Model Structure on Dendroidal Sets 387
9 6 The Absolute Covariant Model Structure ------ 2 -- 396
9 7 The Picard Model Structure -----err2sssoenneneennnnn 404
Part Ill The Homotopy Theory of Simplicial and Dendroidal Spaces
10 Reedy Categories and Diagrams of Spaces ---- - - 423
10 1 Reedy Categories 606 00 c cece eee e eect tent n ee ees 423
10 2 Reedy Fibrations 0 6 00 06 2c eee e eee cece eee eee eens 426
10 3 The Reedy Model Structure -- 20-00 s cence een eee eee 431
10 4 Simplicial Objects and Geometric Realization 435
10 5 Homotopy Colimits 20022 e cece eee eee eee eee ee eee 442
10 6 A Version of Quillen’s Theorem B - -----+--+-0+ eee eee 447
Contents
11 Mapping Spaces and Bousfield Localizations 453
11 1 Mapping Spaces 222 rennen 453
11 2 Common Models for Mapping Spaces 2 460
11 3 Left Bousfield Localizations 000 0 cceeee cesses 467
11 4 Existence of Left Bousfield Localizations 2 469
11 5 Localizable Sets of Morphisms - 000 000 ec eee eee, 474
12 Dendroidal Spaces and oo-Operads 222 nn 481
12 1 Dendroidal Segal Spaces uccecc ee 482
12 2 Complete Dendroidal Segat Spaces 2 2222 49]
12 3 Complete Weak Equivalences 2cen ce 498
12 4 The Tensor Product of Dendroidal Spaces 0 505
12 5 Closed Dendroidal Spaces 22022 509
12 6 Reduced Dendroidal Spaces 2 22222en een 514
12 7 Simplicial Spaces 22unueneee nenn 519
13 Left Fibrations and the Covariant Model Structure 523
13 1 The Covariant Model Structure on Dendroidal Spaces 524
13 2 Simplicial Systems of Model Categories 22222222222222 530
13 3 Homotopy Invariance of the Covariant Model Structure 540
13 4 The Homotopy Theory of Algebras 22 2222222222 544
13 5 Algebras and Left Fibrations 22 2 22 00 549
14 Simplicial Operads and o0-Operads 00 000 000 0 ee eee 55S
14 1 Simplicial Categories with Fixed Objecis 22 22 2222 356
14 2 Equivalences in Simplicial Categories 22 2222222022222 559
14 3 A Model Structure for Simplicial Operads 22 22cm 564
14 4 The Sparse Model Structure 2 222222222 eneeeeennnnenn 567
14 5 Simplicial Operads and Dendroidal Spaces 573
14 6 The Homotopy-Coherent Nerve 00020 2c cece cece eee ees 576
14 7 Operads with a Single Colour 2c2cnceneesnenaneneenn 581
14 8 Algebras for co-Operads and for Simplicial Operads 587
Epilogue 2 0 cc cee eee cece cn nees 59}
References 000 000 cece eee cece cee e pny enes 601 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Heuts, Gijs Moerdijk, Ieke 1958- |
| author_GND | (DE-588)125418211X (DE-588)1067170103 |
| author_facet | Heuts, Gijs Moerdijk, Ieke 1958- |
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| author_sort | Heuts, Gijs |
| author_variant | g h gh i m im |
| building | Verbundindex |
| bvnumber | BV048459190 |
| classification_rvk | SK 300 |
| ctrlnum | (OCoLC)1341403709 (DE-599)BVBBV048459190 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV048459190 |
| illustrated | Not Illustrated |
| index_date | 2024-07-03T20:32:58Z |
| indexdate | 2024-07-10T09:38:43Z |
| institution | BVB |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-033837233 |
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| physical | xx, 612 Seiten Diagramme 24 cm |
| publishDate | 2022 |
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| publishDateSort | 2022 |
| publisher | Springer |
| record_format | marc |
| series2 | Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics |
| spelling | Heuts, Gijs (DE-588)125418211X aut Simplicial and dendroidal homotopy theory Gijs Heuts ; Ieke Moerdijk Cham, Switzerland Springer [2022] xx, 612 Seiten Diagramme 24 cm txt rdacontent n rdamedia nc rdacarrier Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 75 Homotopy theory Moerdijk, Ieke 1958- (DE-588)1067170103 aut Erscheint auch als Online-Ausgabe 978-3-031-10447-3 A Series of Modern Surveys in Mathematics Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge 75 (DE-604)BV000899194 75 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033837233&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Heuts, Gijs Moerdijk, Ieke 1958- Simplicial and dendroidal homotopy theory Homotopy theory |
| title | Simplicial and dendroidal homotopy theory |
| title_auth | Simplicial and dendroidal homotopy theory |
| title_exact_search | Simplicial and dendroidal homotopy theory |
| title_exact_search_txtP | Simplicial and dendroidal homotopy theory |
| title_full | Simplicial and dendroidal homotopy theory Gijs Heuts ; Ieke Moerdijk |
| title_fullStr | Simplicial and dendroidal homotopy theory Gijs Heuts ; Ieke Moerdijk |
| title_full_unstemmed | Simplicial and dendroidal homotopy theory Gijs Heuts ; Ieke Moerdijk |
| title_short | Simplicial and dendroidal homotopy theory |
| title_sort | simplicial and dendroidal homotopy theory |
| topic | Homotopy theory |
| topic_facet | Homotopy theory |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033837233&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000899194 |
| work_keys_str_mv | AT heutsgijs simplicialanddendroidalhomotopytheory AT moerdijkieke simplicialanddendroidalhomotopytheory |