Monoidal categories and functors are ubiquitous in physics as they formalise the notion of systems and processes. It is notably used in many formalisms of quantum field theory, such as functorial field theory and factorisation algebras. I will discuss them in the context of locally covariant quantum field theory as in Rejzner (2016). For the sake of brevity, I will only define strict monoidal categories and functors, even though the examples I will work through later are not strict.
Definition 1.1 (Strict monoidal category). A strict monoidal category is a category equipped with
In a non-strict monoidal category, associativity only holds up to natural isomorphism and the $e$ is a left and right unit up to isomorphism.
A monoidal category generalises the notion of a tensor product on vector spaces. The tensor product takes in two vector spaces or two linear maps over a field and outputs a vector space or linear map over the same field. Similarly, a monoidal product takes in two objects or two morphisms from a category $B$ and outputs another object or morphism from $B$. For vector spaces, $\mathbb{K}$ acts as the unit element as the tensor product of $\mathbb{K}$ with any vector space in $\operatorname{Obj}(\mathbf{Vec}_\mathbb{K})$ is isomorphic to the original space.
In algebraic quantum field theory, algebras of observables are assigned to regions of Minkowski space. Locally covariant quantum field theory generalises this to globally hyperbolic spacetimes.
Definition 1.2 ($\mathbf{Loc}$). A category $\mathbf{Loc}$ has
An embedding $\chi$ is causality preserving if for a causal curve, $\gamma: [a,b]\rightarrow \mathcal{N}$, if its endpoints are in $\chi(\mathcal{M})$, then the entire curve lies in $\chi(\mathcal{M})$.
We need to check whether the composition of morphisms $\chi_1:\mathcal{M}\rightarrow \mathcal{N}$ and $\chi_2:\mathcal{N}\rightarrow \mathcal{O}$, $\chi = \chi_2\circ \chi_1: \mathcal{M}\rightarrow \mathcal{O}$, defines a new morphism. The composition of embeddings is associative and the composition of two orientation preserving isometries is an orientation preserving isometry, so we only need to check that the composition of morphisms preserves causality.
We want to show that for a curve $\gamma: [a,b]\rightarrow \mathcal{O}$ with endpoints in $\chi_2\circ \chi_1 (\mathcal{M})$ the entire curve lies in $\chi_2\circ \chi_1 (\mathcal{M})$. As $\chi_2\in \operatorname{Hom}(\mathbf{Loc})$, we can define a new curve as the preimage of $\gamma(t)$ under $\chi_2$
\[\gamma^\prime=\chi_2^{-1}[\gamma]: [a,b]\rightarrow \mathcal{N}.\]As the $\gamma(a),\gamma(b)\in\chi_2\circ \chi_1 (\mathcal{M})$, the endpoints of $\gamma^\prime$ are in $\chi_1 (\mathcal{M})$. Because $\chi_1\in \operatorname{Hom}(\mathbf{Loc})$, we must have $\gamma^\prime(t)\in \chi_1(\mathcal{M})$ for all $t\in (a,b)$. This implies that $\gamma(t)\in \chi_2\circ\chi_1(\mathcal{M})$ for all $t\in (a,b)$. So $\chi = \chi_2\circ\chi_1\in \operatorname{Hom}(\mathbf{Loc})$.
We can extend $\mathbf{Loc}$ to a monoidal category $\mathbf{Loc}^\sqcup$ by taking the disjoint union as the monoidal product and the empty set as the unit element. The disjoint union is associative up to natural isomorphism
\[\mathcal{M}_1\sqcup (\mathcal{M}_2\sqcup \mathcal{M}_3)\cong( \mathcal{M}_1\sqcup \mathcal{M}_2)\sqcup \mathcal{M}_3\quad \forall \mathcal{M}_1, \mathcal{M}_2, \mathcal{M}_3\in \operatorname{Obj}(\mathbf{Loc}),\]and the disjoint union of the null set provides a left and right unit up to isomorphism
\[\emptyset\sqcup \mathcal{M} \cong \mathcal{M} \cong \mathcal{M} \sqcup\emptyset.\]As the two regions in the spacetime $\mathcal{M}_1\sqcup \mathcal{M}_2$ are disjoint, it follows that they are spacelike separated. Morphisms in $\mathbf{Loc}$ are causality preserving so for an embedding $\chi:\mathcal{M}_1\sqcup \mathcal{M}_2\rightarrow \mathcal{M}$, the images of these regions $\chi(\mathcal{M}_1)$, $\chi(\mathcal{M}_2)$ must also be spacelike in $\mathcal{M}$.
Generally, the algebra of observables for a given region will be a $\star$-algebra (potentially with extra structure).
Definition 1.3 ($\mathbf{Obs}$). The category of observables has
The morphisms are algebra homomorphisms, $\varphi:\mathcal{A}\rightarrow \mathcal{B}$, with the additional properties, $\varphi(\mathbb{1}_\mathcal{A}) = \mathbb{1}_\mathcal{B}$ and $\varphi(A^\star) = \varphi(A)^\star$.
This is easily equipped with the familiar tensor product on vector spaces and its unit element is also the complex numbers. (This requires more care when dealing with $C^\star$-algebras.)
The final notion we require is a functor between these categories, $\mathfrak{A}:\mathbf{Loc}\rightarrow \mathbf{Obs}$, that respects their monoidal structure.
Definition 1.4 (Strict monoidal functor). A functor between monoidal categories that respects their monoidal structures. For the functor $\mathfrak{A}$ defined above, this is the requirement that
\[\mathfrak{A}(\mathcal{M}\sqcup \mathcal{M}^\prime)=\mathfrak{A}(\mathcal{M})\otimes\mathfrak{A}(\mathcal{M}^\prime),\] \[\mathfrak{A}_{\chi\sqcup\chi^\prime}=\mathfrak{A}_\chi \otimes \mathfrak{A}_{\chi^\prime},\] \[\mathfrak{A}(\emptyset)=\mathbb{C}.\]This functor is a model of a locally covariant quantum field theory.
To do: create figures, , explain the $\star$-structure on the $\mathbf{Loc}$ side, add section on quantum fields in this model.