$\mathbb{R}$-Vect is cototal?

Question

Let $\mathbb{R}$-vect denote the category of $\mathbb{R}$-vector space and linear maps. Does this form a cototal category?

Answer 1

Sure. This is true for any complete well-powered category (or any complete category admitting intersections of all families of subobjects) with a cogenerator, which $\mathbb R$ is in our case. See the nLab article on total categories for details. Roughly, it's the requirement for a cogenerator which makes cototal categories empirically rarer than total ones.