Using the simple example of a free scalar field, I will illustrate the axiomatic formulation of locally covariant quantum field theory (LCQFT). With this framework in mind, the case of certain Abelian gauge theories will be discussed. In particular, I will present a model for free electromagnetism (and higher analogues) and its variations in terms of doubled and self-dual fields. As it turns out, despite being among the easiest examples of a genuine gauge theory, these models do not fulfil the locality axiom of LCQFT, i.e. the requirement that causal embeddings between spacetimes induce injective morphisms between the associated (C*-)algebras of observables. This observation motivates us to look for a softer version of LCQFT that fits gauge theories too. For this purpose, we propose an homotopy theoretic approach to the problem.
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