Functors over a base as generalized multicategories [mde-VRYH]

\(T\)-monoids for \(T\) the free presheaf monad extended to \(\mathbf {Span}(\mathbf {Set}^{S_{\text {obj}}})\) are functors over \(S\).

Let \(X : S_{\text {obj}} \to \mathbf {Set}\) be a family of sets, and consider a \(T\)-monoid with carrier \(X\). We define a functor \(P : \mathbb {X} \to S\) by unravelling the data of this \(T\)-monoid.

• The elements of \(X_s\) are the objects of the fibre of \(P\) over \(s\).
• The underlying span of the \(T\)-monoid, \(X \xleftarrow {D} M \xrightarrow {C} TX\) comprises \(S_{\text {obj}}\)-indexed families of functions, \[D_s : M_s \to X_s\] and \[C_s : M_s \to \sum _{t \in S} X_{t} \times S(s,t)\] where \(M_s\) is the set of morphisms of \(\mathbb {X}\) with domain in \(X_s\), with \(D_s\) giving this domain, and where \(C_s\) gives for every morphism with domain in \(X_s\) its codomain and the morphism in \(S\) to which \(P\) maps it. So far this specifies a graph over \(S\).
• The unit cell of the \(T\)-monoid amounts to a family of functions, \[u_s : X_s \to M_s\] assigning to each object \(x\) in the fibre over \(s\) an endomorphism \(i : x \to x\) such that its image under \(P\) is the identity on \(s\), i.e. candidate identity morphisms.
• The multiplication cell of the \(T\)-monoid amounts to a family of functions, \[m_s : (M \times _{TX} TM)_s \to M_s\] assigning to each pair of composable morphisms in \(\mathbb {X}\) with domain of the first morphism over \(s\) their composite, in a functorial manner.
• finally the monoid laws witness that the candidate identities and composites indeed satisfy unitality and associativity.