turkmath.org

We first discuss fibration category structures induced by enrichments in symmetric monoidal categories that are also fibration categories. Then we discuss extension of these structures to equivariant context and show that for a group G and a symmetric monoidal category V which is also a fibration category, under mild conditions the category of G-objects in a V-enriched category admits a nontrivial fibration category structure. Lastly we discuss some examples of such fibration categories and applications.