The n-Category Café

For $V$ a (finite dimensional) vector space, I write $FinSet_V$ for the category of sets over $V$. An object is a set $S$ with a map to $V$, $$S\to V$$ and a morphism is a commuting triangle $$\array{ S &\to& S' \\ & \searrow \swarrow \\ & V } \,.$$ The $V$-cardinality operation is a monoidal map $$|\cdot| : (FinSet_V, \oplus) \to (V, +)$$ sending a finite set labeled by $V$ to the sum of the labels of its elements.

Let $\mathbf{B} End(V)$ be the category whose single object is the vector space $V$ and whose space of morphisms is $End(V)$, with composition of morphisms being the composition of these endomorphisms.

Take $C$ to be any finite category whose category algebra we want to consider. Form the cartesian product category $$C \times \mathbf{B} End(V)$$ and consider finite sets over the collection of morphisms: $$\array{ S \\ \downarrow \\ Mor(C \times \mathbf{B} End(V)) } \,.$$ Notice that for each fixed morphisms of $C$ this is an element in $FinSet_{End(V)}$. Under the $End(V)$-cardinality it is therefore an $End(V)$-valued function on $Mor(C)$. If $V = \mathbb{C}$ (and assuming we are talking about complex vector spaces), then this is a complex function on $Mor(C)$.

Given two such sets over $Mor(C \times \mathbf{B}End(V))$ $$\array{ S &&&& S' \\ \downarrow &&&& \downarrow \\ Mor(C \times \mathbf{B} End(V)) &&&& Mor(C \times \mathbf{B} End(V)) }$$ representing two $End(V)$-valued functions on $Mor(C)$ we take the correspondence space given by the composition operation in $C \times \mathbf{B}End(V)$

$$\array{ S && Mor(C \times \mathbf{B} End(V)) {}_t \times_s Mor(C \times \mathbf{B} End(V)) && S' \\ \downarrow & {}^{d_0}\swarrow &\downarrow^\circ& \searrow^{d_1} & \downarrow \\ Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) } \,,$$ pull back both sets $S$ and $S'$ to the total space and take fiberwise cartesian product there to get the new set $d_0^* S \otimes d_1^* S'$ $$\array{ && d_0^* S \otimes d_1^* S' \\ && \downarrow \\ && Mor(C \times \mathbf{B} End(V)) {}_t \times_s Mor(C \times \mathbf{B} End(V)) && \\ & {}^{d_0}\swarrow &\downarrow^\circ& \searrow^{d_1} & \\ Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) }$$ and then push that down along the composition map $\circ$ $$\array{ && Mor(C \times \mathbf{B} End(V)) {}_t \times_s Mor(C \times \mathbf{B} End(V)) \\ & {}^{d_0}\swarrow &\downarrow^\circ& \searrow^{d_1} & \\ Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) && Mor(C \times \mathbf{B} End(V)) \\ && \uparrow \\ && \int_\circ d_0^* S \otimes d_1^* S' } \,.$$ Under $End(V)$-cardinality, the resulting finite set $\int_\circ d_0^* S \otimes d_1^* S'$ is the category algebra product of the two original $End(V)$-valued functions on $Mor(C)$.

We can also discuss the action of this category algebra on $V$-valued functions on $Obj(C)$ in a similar manner:

For $A$ a monoid write $\mathbf{B}A$ for the corresponding one-object category. A representation of $A$ on $V$ is a functor $$\rho : \mathbf{B}A \to Vect$$ such that $$\rho : (\bullet \stackrel{a}{\to} \bullet) \mapsto (V \stackrel{\rho(a)}{\to} V) \,.$$ We write $$V//A$$ for the corresponding action category, being the pullback $$\array{ V//A &\to& Vect_* &\to& Set_* \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B}A &\stackrel{\rho}{\to}& Vect &\to& Set }$$ which has $V$ as its space of objects and one morphism of the form $$v \stackrel{a}{\to} \rho(a)(v)$$ for each $v \in V$ and each $a \in A$, with the obvious composition coming from the product in $A$.

For the discussion of category algebras, consider again the monoid $A = End(V)$ and hence the action category $V//End(V)$. Notice that every $End(V)$-valued set over $Mor(C \times \mathbf{B}End(V))$ pulls back along the canonical projection $$p : V//A \to \mathbf{B}A$$ to a finite set over $Mor(C \times V//End(V))$.

So consider a $V$-valued set over $Obj(C)$ in the form of a finite set $$\array{ S \\ \downarrow \\ Obj(C \times V//End(V)) ) = Obj(C)\times V }$$ and then the correspondence space of the category $C$ $$\array{ && p^* S' \\ && \downarrow \\ S && Mor(C \times V//End(V)) \\ \downarrow & {}^s\swarrow && \searrow^t \\ Obj(C)\times V &&&& Obj(C)\times V } \,,$$

where $p^* S'$ is the pullback along the above $p$ of an $End(V)$-valued set over $Mor(C)$ representing, by the previous discussion, an element in the category algebra of $C$ (with coefficients in $V$).

Then pull back $S$ along the source map to $Mor(C \times V//End(V))$, tensor there with $p^* S'$

$$\array{ && s^*S \otimes p^* S' \\ && \downarrow \\ && Mor(C \times V//End(V)) \\ & {}^s\swarrow && \searrow^t \\ Obj(C)\times V &&&& Obj(C)\times V } \,,$$

and then push down along the target map

$$\array{ && Mor(C \times V//End(V)) && \int_t s^* S \otimes p^* S' \\ & {}^s\swarrow && \searrow^t & \downarrow \\ Obj(C)\times V &&&& Obj(C)\times V }$$

to get a $V$-valued set over $Obj(C)$. Under $V$-cardinality, this is the image of the original $V$-valued function represented by $S$ under the action of the category algebra element represented by $S'$.

The reasoning needed for seeing this is precisely the one used for the disucssion of the path integral in this context.