We present a general approach to homotopy limits via homotopy ends, valid in all combinatorial model categories, without the need for simplicial powerings or Quillen 2-adjunctions. Our main tool is the observation that the end functor $\smash { \int_{\Gamma} \colon \mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma} \to \mathscr{C} } $ is right Quillen if the diagram category $ \mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma} $ is equipped with any of the following model structures: $ \smash{ (\mathscr{C}^{\Gamma^{\mathrm{op}}}_{\mathrm{Proj}})^{\Gamma}_{\mathrm{Inj}} } $, $ \smash{ (\mathscr{C}^{\Gamma}_{\mathrm{Proj}})^{\Gamma^{\mathrm{op}}}_{\mathrm{Inj}} } $, or $ \smash { \mathscr{C}^{\Gamma^{\mathrm{op}}\times\Gamma }_{\mathrm{Reedy}} } $. 查看全文>>