Approximate Injectivity

Abstract

In a locally λ-presentable category, with λ a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are λ-presentable, are known to be characterized by their closure under products, λ-directed colimits and λ-pure subobjects. Replacing the strict commutativity of diagrams by "commutativity up to ε", this paper provides an "approximate version" of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed ε-generalizations of the notion of λ-purity. The categorical theory is being applied to the locally ℵ 1 -presentable category of Banach spaces and their linear operators of norm at most 1, culminating in a largely categorical proof for the existence of the so-called Gurarii Banach space.

Enriched purity and presentability in Banach spaces

“…We also show that pure morphisms have the same model-theoretical meaning as pure morphisms in accessible categories when we use the logic of positive bounded formulas of [14,15]. This complements the results of [31] where purity in metric enriched categories was introduced. Using purity, we Date: June 11, 2022.…”

“…A functor F : K → L between CMet-enriched categories is enriched if the mapping K(A, B) → L(F A, F B) is nonexpansive for all objects A and B in K. An adjunction U ⊢ F is enriched if L(F K, L) ∼ = K(K, UL) is an isomorphism of metric spaces for all objects K in K and L in L. Consult [3] for the concept of a locally presentable category and [16] for the theory of enriched categories. Enriched categories over CMet were studied in [31] and [4] from where we recall the following concepts.…”

“…Our aim is to place this approach into the framework of the theory of locally presentable and accessible categories. The first steps in this direction were done in [31] and [4]. Accessible categories were introduced by M. Makkai and R. Paré [27] as a foundation of categorical model theory.…”

Are Banach spaces monadic?

“…The category CMet has many deficiencies and it is natural to replace it by the category CMet ∞ of generalized complete metric spaces by allowing distances to be ∞ while keeping all other requirements, as well as the type of morphisms (see, e.g., [16]). In the same way, we can generalize Banach spaces by allowing norms to be ∞.…”

Metric enrichment, finite generation, and the path comonad

“…Such metric-flavored category-theoretic considerations are by now pervasive in the literature: in discussing universal (Gurarii) Banach spaces [20,21], or universal operators thereon [14], or more general issues of approximate embeddability [25,3]; these are only a handful of examples, each with its own extensive cited literature.…”