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A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
10
votes
Model Structure/Homotopy Pushouts in topological monoids?
It's easy to describe the group completion of a pushout, up to homotopy. If $Y\leftarrow X \rightarrow Z$ is the diagram of topological monoids, then the group completion of the pushout is equivalent …