Let $\mathcal{C}$ denote a category with a zero object. From the following definition of a norm of an object, where, if $X$ is in $\mathrm{Ob}(\mathcal{C})$, then $$ \text{N}(X) := \prod_{[P]\in \text{P}(\mathcal{C})}|\mathrm{Hom}(X, P)|, $$ where $\mathrm{P}(\mathcal{C})$ denotes the isomorphism classes of all finite simple objects, where finite simple object means $|\mathrm{Hom}(P,P)|$ is finite for a simple object $P$. We can extend this notion to say a non-zero object $X$ is finite if $\mathrm{N}(X)$ is finite. As demonstrated in Z. Goldthorpe's, if we define the categorical zeta function of our category $\mathcal{C}$ as
$$\zeta_\mathcal{C}(s) = \prod_{[P] \in \mathrm{P}(\mathcal{C})} \frac{1}{1-\text{N}(P)^{-s}},$$ then this definition has the advantage of not only agreeing with Kurokawa's categorical zeta function $\zeta_\mathcal{C}^{\text{Ku}}(s)$in [1] but also satisfies $$ \prod_{[P]\in \text{P}(\mathcal{C})}\frac1{1-\text{N}(P)^{-s}} = \sum_{[X]\in \text{M}(\mathcal{C})}\frac1{\text{N}(X)^s} $$ whenever $\mathcal{C}$ is a category with a zero object with finite coproducts such that the finite simple objects satisfy Schur's lemma, and $\text{M}(\mathcal{C})$ denotes the isomorphism classes of finite coproducts of finite simple objects$.^1$
Examples: Per Kurokawa, if $\mathcal{C}$ denotes the category of $\mathbb{Z}$-modules, we have $\zeta_{\mathcal{C}}(s) = \zeta(s)$, the Riemann zeta function. Generalizing this to $O_K$-modules, where $O_K$ denotes the ring of integers of a number field $K$, we have $\zeta_{\mathcal{C}}(s) = \zeta_K(s)$, the Dedekind zeta function.
Now, in [2], there is another definition of a categorical zeta function, namely Noguchi's categorical zeta function, defined as
$$\zeta_\mathcal{C}^{\text{No}}(t) = \exp\left(\sum_{m=1}^\infty \frac{\text{N}_m(\mathcal{C})}{m} t^m\right),$$ where $\mathcal{C}$ is a finite category, i.e., a category having finitely many objects and morphisms and $\text{N}_m(\mathcal{C}) = |\{ (X_0 \xrightarrow{f_1} X_1 \xrightarrow{f_1} \cdots \xrightarrow{f_m} X_m) \; \text{in} \; \mathcal{C} \}|$. Interestingly, K. Noguchi's remarks $\zeta_\mathcal{C}^{\text{No}}(t)$ is different from $\zeta_\mathcal{C}^\text{Ku}(s)$ since Kurokawa's allows a choice for large categories.
Inspired by the construction of global objects from local objects, we will define the category $\mathcal{C}$ with a removed object $X$, denoted $\mathcal{C} \smallsetminus X$ (abuse of notation), where $\mathrm{Ob}(C \smallsetminus X) = \mathrm{Ob}(\mathcal{C})$ and $\mathrm{Hom}(\mathcal{C} \smallsetminus X)$ are all morphisms in $\mathrm{Hom}(\mathcal{C})$, except for our morphisms in $\text{Hom}(X_1,X)$ and in $\text{Hom}(X,X_2)$ for any $X_1,X_2\in\text{Ob}(\mathbf{C})$--this is, in fact, a category. We will similarly define the localization of our category $\mathcal{C}_X := \mathcal{C}[\mathrm{Hom}(\mathcal{\mathcal{C}} \smallsetminus X)^{-1}]$ as the local category of $\mathcal{C}$ at $X$.
Question: Is it true, for any category with a zero object, we have $$\zeta_\mathcal{C}(s) = \prod \zeta^{\text{No}}_{\mathcal{C}_P}(s)$$ where the product is over $\mathcal{C}_P$, which is a representative of equivalence classes of categories $[\mathcal{C}_P]$ whose objects lie in $\mathrm{FinCat}$, and $P$ is a representative of our isomorphism class $[P] \in \mathrm{P}(\mathcal{C})$?
References:
1.) N. Kurokawa. Zeta Functions of Categories. Proc. Japan Acad. Vol 72A No. 10, 221-222 (1996).
2.) Kazunori Noguchi. The zeta function of a finite category. Doc. Math. 18, 1243–1274 (2013).
Footnotes:
1.) Technically, the norm is on the equivalence class of our object $X$; however, this does not pose a problem, since the norm $\mathrm{N}([X])$ is well-defined under equivalence. We may choose a representative of our isomorphism class $[X]$ and therefore favor the notation $\mathrm{N}([X]) = \mathrm{N}(X)$.
