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We consider several cases of non-existence theorems for functors. For example, there are no nontrivial functors from the category of sets, (or the category of groups, or vector spaces) to any small category. See 2.3. Another kind of nonexistence is that of (co-)augmented functors. For example, every augmented functor from groups to abelian groups, is trivial, i.e. has a trivial augmentation map. Every surjective co-augmented functor from groups to perfect groups or to free groups is also trivial.

The knowledge that data lies close to a particular submanifold of the ambient Euclidean space may be useful in a number of ways. For instance, one may want to automatically mark any point far away from the submanifold as an outlier or to use the geometry to come up with a better distance metric. Manifold learning problems are often posed in a very high dimension, e.g. for spaces of images or spaces of words. Today, with deep representation learning on the rise in areas such as computer vision and natural language processing, many problems of this kind may be transformed into problems of moderately high dimension, typically of the order of hundreds. Motivated by this, we propose a manifold learning technique suitable for moderately high dimension and large datasets. The manifold is learned from the training data in the form of an intersection of quadric hypersurfaces -- simple but expressive objects. At test time, this manifold can be used to introduce a computationally efficient outlier score for arbitrary new data points and to improve a given similarity metric by incorporating the learned geometric structure into it.