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Monday, September 21, 2020

**Abstract:** For any quasicategory $X$ with objects $w,x$, $\textrm{Hom}(w,x)$ is a Kan complex. Given any edge, $f: x \rightarrow y$ in $X$, we can construct a functor $f_{*}: \textrm{Hom}(w,x) \rightarrow \textrm{Hom}(w,y)$ but this construction is such that for a pair of composable edges, $f: x \rightarrow y$ and $g: y \rightarrow z$, the lifts $g_{*}f_{*}$ and $(gf)_{*}$ are not necessarily equal, only homotopic. Consequently, $\textrm{Hom}(w,-)$ does not induce a functor from $X$ into the category of Kan complexes. We will show how we can `straighten' a homotopy coherent functor (i.e. a left fibration) such as $\textrm{Hom}(w,-)$ into an actual functor between infinity categories. Please email vb8 at illinois dot edu for the zoom details.