### Homotopy Theory Seminar

Thursday, October 20, 2022 - 3:30pm

#### Maru Sarazola

Johns Hopkins University

A natural way to construct a model structure on a category $\mathcal{C}$ is to right induce it through a right adjoint $R\colon\mathcal{C}\to\mathcal{M}$ to a known model structure $\mathcal{M}$. This process defines the fibrations and weak equivalences in $\mathcal{C} as the morphisms whose image under $R$ is one such map in $\mathcal{M}$. Depending on the context, however, this may prove too restrictive. For example, one may be working in a setting where there is a desired class of “fibrant objects” in mind for $\mathcal{C}$, and where the best one can hope for is for these well-behaved classes of fibrations and weak equivalences to hold only between fibrant objects. This leads to what we call a semi-right induced model structure. This talk will present the idea of semi-right induced model categories. We will show an example where the right induced model structure does not exist, but the semi version does, and explore how this process may be used to define model structures on categories internal to n-categories. Based on work in progress together with Lyne Moser and Paula Verdugo.