# Effective homology for homotopy colimit and cofibrant replacement

Archivum Mathematicum (2014)

• Volume: 050, Issue: 5, page 273-286
• ISSN: 0044-8753

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## Abstract

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We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X:ℐ\to \text{sSet}$ such that each simplicial set $X\left(i\right)$ has effective homology, we present an algorithm computing the homotopy colimit $\text{hocolim}\phantom{\rule{0.166667em}{0ex}}X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement ${X}^{\text{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.

## How to cite

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Filakovský, Marek. "Effective homology for homotopy colimit and cofibrant replacement." Archivum Mathematicum 050.5 (2014): 273-286. <http://eudml.org/doc/262212>.

@article{Filakovský2014,
abstract = {We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal \{I\}\rightarrow \mbox\{sSet\}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mbox\{hocolim\}\,X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^\{\mbox\{cof\}\}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.},
author = {Filakovský, Marek},
journal = {Archivum Mathematicum},
keywords = {homotopy colimit; cofibrant replacement; effective homology; equivariant; homotopy colimit; cofibrant replacement; effective homology; equivariant},
language = {eng},
number = {5},
pages = {273-286},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Effective homology for homotopy colimit and cofibrant replacement},
url = {http://eudml.org/doc/262212},
volume = {050},
year = {2014},
}

TY - JOUR
AU - Filakovský, Marek
TI - Effective homology for homotopy colimit and cofibrant replacement
JO - Archivum Mathematicum
PY - 2014
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 050
IS - 5
SP - 273
EP - 286
AB - We extend the notion of simplicial set with effective homology presented in [22] to diagrams of simplicial sets. Further, for a given finite diagram of simplicial sets $X \colon \mathcal {I}\rightarrow \mbox{sSet}$ such that each simplicial set $X(i)$ has effective homology, we present an algorithm computing the homotopy colimit $\mbox{hocolim}\,X$ as a simplicial set with effective homology. We also give an algorithm computing the cofibrant replacement $X^{\mbox{cof}}$ of $X$ as a diagram with effective homology. This is applied to computing of equivariant cohomology operations.
LA - eng
KW - homotopy colimit; cofibrant replacement; effective homology; equivariant; homotopy colimit; cofibrant replacement; effective homology; equivariant
UR - http://eudml.org/doc/262212
ER -

## References

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