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# Homology, Homotopy and Applications

## Volume 19 (2017)

### Number 2

### Extending homotopy theories across adjunctions

Pages: 89 – 110

DOI: http://dx.doi.org/10.4310/HHA.2017.v19.n2.a6

#### Authors

#### Abstract

Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $\Gamma$-categories.

#### Keywords

lax map, strict map, $2$-monad

#### 2010 Mathematics Subject Classification

18A25, 18C20, 18D50, 19D23, 55U35

Paper received on 4 October 2016.