Let F: C → Vect be a functor from a category C to vector spaces over a normed field. A functorial semi-norm on F is a factorization of F over the forgetful functor snVect → Vect, where snVect denotes the corresponding category of semi-normed vector spaces.
Functorial semi-norms, in particular the ℓ¹-semi-norm, on singular homology were introduced by Gromov in his study of simplicial volume of manifolds. The latter is a homotopy invariant that, roughly speaking, measures the complexity of the fundamental class of an oriented closed connected manifold.

In the present thesis, we investigate three aspects of functorial semi-norms:

Universal functorial semi-norms (joint work with Clara Löh).
For a fixed functor, we define a relation among all its functorial semi-norms, whose minimal elements we call universal. We then prove certain existence results of such universal functorial semi-norms.

Inflexibility from a computational perspective.
Crowley and Löh established a bidirectional correspondence between functorial semi-norms on singular homology and so-called inflexible manifolds. The construction of such manifolds is based on the construction and study of certain differential graded algebras, which are purely algebraic objects. As such they are very amenable to computations, not only by humans but also via computer programs. We present fragments of a software that facilitates such computations, some new examples that we found in this way, and two results about algorithmic decidability.

Excisive approximation of ℓ¹-homology.
It is a well-known fact, that ℓ¹-homology does not satisfy the excision axiom in the sense of Eilenberg and Steenrod. On the other hand, the fact that singular homology satisfies excision is already visible at the level of the singular chain complex functor, namely the latter is excisive in the sense of Goodwillie calculus. The latter, however, also provides the framework for constructing a universal (or best) excisive approximation to a given functor. We apply this theory to the ℓ¹-chain complex functor and show that its excisive approximation vanishes.
In appendices, we include a proof of the fact that the singular chain complex functor is excisive in the sense of Goodwillie calculus, and we relate this abstract form of excision to the classical one in the form of Mayer-Vietoris sequences.

Sei F: C → Vect ein Funktor von einer Kategorie C in eine Kategorie von Vektorräumen über einem normierten Körper. Eine funktorielle Seminorm auf F ist eine Faktorisierung von F über den Vergissfunktor snVect → Vect, wobei snVect die zugehörige Kategorie von seminormierten Vektorräumen bezeichne.
Funktorielle Seminormen, insbesondere die ℓ¹-Seminorm, auf singulärer Homologie wurden von Gromov bei seinem Studium des simplizialen Volumens von Mannigfaltigkeiten eingeführt. Letzteres ist eine Homotopieinvariante, welche grob gesagt die Komplexität der Fundamentalklasse einer orientierten, geschlossenen, zusammenhängenden Mannigfaltigkeit misst.

In der vorliegenden Arbeit untersuchen wir drei Aspekte funktorieller Seminormen:
Universalität funktorieller Seminormen (gemeinsame Arbeit mit Clara Löh).
Inflexibilität aus dem Blickwinkel computergestützter Berechnungen.
Ausschneidung-erfüllende Annäherung an ℓ¹-Homologie.