Path spaces of pushouts
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908478862262272 |
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| author | Wärn, David |
| author_facet | Wärn, David |
| contents | Given a span of spaces, one can form the homotopy pushout and then take the homotopy pullback of the resulting cospan. We give a concrete description of this pullback as the colimit of a sequence of approximations, using what we call the zigzag construction. We also obtain a description of loop spaces of homotopy pushouts. Using the zigzag construction, we reproduce generalisations of the Blakers-Massey theorem and fundamental results from Bass-Serre theory. We also describe the loop space of a wedge and show that it splits after suspension. Our construction can be interpreted in a large class of $(\infty,1)$-categories and in homotopy type theory, where it resolves the long-standing open problem of showing that a pushout of 0-types is 1-truncated. The zigzag construction is closely related to the James construction, but works in greater generality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_12339 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Path spaces of pushouts Wärn, David Algebraic Topology Category Theory Given a span of spaces, one can form the homotopy pushout and then take the homotopy pullback of the resulting cospan. We give a concrete description of this pullback as the colimit of a sequence of approximations, using what we call the zigzag construction. We also obtain a description of loop spaces of homotopy pushouts. Using the zigzag construction, we reproduce generalisations of the Blakers-Massey theorem and fundamental results from Bass-Serre theory. We also describe the loop space of a wedge and show that it splits after suspension. Our construction can be interpreted in a large class of $(\infty,1)$-categories and in homotopy type theory, where it resolves the long-standing open problem of showing that a pushout of 0-types is 1-truncated. The zigzag construction is closely related to the James construction, but works in greater generality. |
| title | Path spaces of pushouts |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2402.12339 |