ora-0024
1.12 Homotopy limits, colimits, and derived Kan extensions
Ordinary limits and colimits are not generally invariant under objectwise weak equivalence of diagrams. Homotopy limits and colimits correct them so that weakly equivalent input diagrams have equivalent universal outputs. Riehl develops these constructions through enriched weighted (co)limits and derived functors [ Riehl , 2014 ] .
In a localized decision target \(\mathcal D_\infty \), define
The superscript \(h\) records Kan extension in the homotopy-coherent semantics. In a compatible model presentation these are total derived Kan extensions.
Left Kan extension is a left adjoint and therefore preserves homotopy colimits. Right Kan extension is computed pointwise by homotopy limits. The asymmetry becomes decisive online: incremental candidate generation commutes with filtered revelation under broad conditions, whereas right-stage consistency requires an interchange between filtered colimits and the relevant limits.