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Jade Master (Jul 26 2022 at 12:22):Does anyone know about the theory of V-enrichment when V is a symmetric monoidal category which is not closed? The reason is that I am interested in -enriched categories with the cartesian monoidal structure. This monoidal structure is not closed (it is the usual cocartesian structure + in Set) because it does not distribute over coproducts (products in Set). I know that this means that will not be self-enriched. but we can certainly still define a -category. Is there any way to talk about weighted limits/colimits in this setting?
Mike Shulman (Jul 26 2022 at 15:03):Yes, weighted limits and colimits can be defined: you just stipulate as part of their definition that the hom-object into the limit has the universal property that the "type of weighted cones" would have if it existed. In fact weighted limits and colimits can be defined in any virtual equipment. This definition is implicitly in 8.1 of Enriched indexed categories, though not phrased there in fully generality.