A Category of Categorifications over a Set

Categorifications over a Base and the Monoid Ring ConstructionAbstract1. Naive Decategorification2. Categorifications over a Base3. The Basic Example: $\mathbb N$4. Functorial Base Transformation under Decategorification5. The Finite-Support $M$-Grading Functor6. Day Convolution and the Monoid Semiring $\mathbb N[M]$7. $M$-Graded Vector Spaces and the Monoid Ring8. Linearization and the Lifted Free Vector Space Functor9. The Relative Viewpoint Revisited10. Summary

 

Categorifications over a Base and the Monoid Ring Construction

Abstract

We introduce a relative viewpoint on naive decategorification. Instead of only asking whether a category decategorifies to a fixed set $B$$B$, we consider categories equipped with a chosen map from their set of object-isomorphism classes to $B$$B$. This leads to the comma category

$${\mathsf{Cat}}_{/B}^{\mathrm{decat}}=(\mathrm{Decat}\downarrow B).$$

Within this framework, the categories $\mathsf{FinSet}$$\mathsf{FinSet}$ and ${\mathsf{FinVect}}_k$$\mathsf{FinVect}_k$ appear as two categorifications of $\mathbb{N}$$\mathbb N$, and the free vector space functor $k[-]$$k[-]$ becomes a morphism between these categorifications.

We then isolate a general mechanism: an endofunctor $T$$T$ on categories induces a transformation between categories of categorifications over bases only when its effect on decategorification is controlled by a compatible set-level functor $S$$S$. The finite-support $M$$M$-grading construction

$$\mathcal{C}\mapsto {\mathrm{Fun}}_{\mathrm{fs}}(D(M),\mathcal{C})$$

is an important example. It sends categorifications over $\mathbb{N}$$\mathbb N$ to categorifications over $\mathbb{N}[M]$$\mathbb N[M]$. With Day convolution, finite-support $M$$M$-graded finite sets categorify the monoid semiring $\mathbb{N}[M]$$\mathbb N[M]$, while finite-support $M$$M$-graded finite-dimensional vector spaces have naive decategorification $\mathbb{N}[M]$$\mathbb N[M]$ and Grothendieck group $\mathbb{Z}[M]$$\mathbb Z[M]$. Thus the monoid ring $\mathbb{Z}[M]$$\mathbb Z[M]$ is obtained by first passing to $M$$M$-graded categorifications and then applying $K_0$$K_0$.

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1. Naive Decategorification

Let $\mathsf{Cat}$$\mathsf{Cat}$ denote a suitable category of essentially small categories. Define the naive decategorification functor

$$\mathrm{Decat}:\mathsf{Cat}\to \mathsf{Set}$$

by

$$\mathrm{Decat}(\mathcal{C})=\mathrm{Obj}(\mathcal{C})/\cong .$$

Equivalently,

$$\mathrm{Decat}(\mathcal{C})={\pi }_0({\mathcal{C}}^{\simeq }),$$

where ${\mathcal{C}}^{\simeq }$$\mathcal C^{\simeq}$ is the core groupoid of $\mathcal{C}$$\mathcal C$, obtained by keeping all objects of $\mathcal{C}$$\mathcal C$ but only the isomorphisms.

If $F:\mathcal{C}\to \mathcal{D}$$F:\mathcal C\to\mathcal D$ is a functor, then $F$$F$ induces a function

$$\mathrm{Decat}(F):\mathrm{Decat}(\mathcal{C})\to \mathrm{Decat}(\mathcal{D}),[X]\mapsto [F(X)].$$

This is well-defined because functors preserve isomorphisms. Hence $\mathrm{Decat}$$\operatorname{Decat}$ is a functor from categories to sets.

This is the most elementary form of decategorification: one forgets all non-invertible morphisms, then collapses isomorphic objects to the same element. Later, when additive or exact structures are present, one may refine this by passing from ${\pi }_0$$\pi_0$ to a Grothendieck group $K_0$$K_0$. But the starting point of this paper is the naive invariant $\mathrm{Obj}(\mathcal{C})/\cong$$\operatorname{Obj}(\mathcal C)/\cong$.

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2. Categorifications over a Base

Fix a set $B$$B$. We define the category of categorifications over $B$$B$ to be the comma category

$${\mathsf{Cat}}_{/B}^{\mathrm{decat}}=(\mathrm{Decat}\downarrow B).$$

An object of ${\mathsf{Cat}}_{/B}^{\mathrm{decat}}$$\mathsf{Cat}^{\operatorname{decat}}_{/B}$ is a pair

$$(\mathcal{C},p),$$

where $\mathcal{C}$$\mathcal C$ is a category and

$$p:\mathrm{Decat}(\mathcal{C})\to B$$

is a function. Thus each object of $\mathcal{C}$$\mathcal C$, up to isomorphism, is assigned a decategorified value in $B$$B$.

A morphism

$$F:(\mathcal{C},p)\to (\mathcal{D},q)$$

is a functor

$$F:\mathcal{C}\to \mathcal{D}$$

such that

$$q\circ \mathrm{Decat}(F)=p.$$

Equivalently, for every object $X\in \mathcal{C}$$X\in\mathcal C$, one has

$$q([F(X)])=p([X]).$$

Thus morphisms in ${\mathsf{Cat}}_{/B}^{\mathrm{decat}}$$\mathsf{Cat}^{\operatorname{decat}}_{/B}$ are functors that preserve the assigned decategorified value.

This definition is intentionally relative. It does not require

$$\mathrm{Decat}(\mathcal{C})\cong B.$$

It only requires a chosen map

$$\mathrm{Decat}(\mathcal{C})\to B.$$

This is useful because many natural categories do not decategorify onto the entire intended base. For example, finite-dimensional vector spaces naturally produce nonnegative dimensions, so their naive decategorification gives $\mathbb{N}$$\mathbb N$, while the corresponding Grothendieck group gives $\mathbb{Z}$$\mathbb Z$.

There is nevertheless a stronger notion. Define

$${\mathsf{Cat}}_B^{\mathrm{cat}}\subset {\mathsf{Cat}}_{/B}^{\mathrm{decat}}$$

to be the full subcategory whose objects are pairs $(\mathcal{C},p)$$(\mathcal C,p)$ such that

$$p:\mathrm{Decat}(\mathcal{C})\to B$$

is a bijection. In that case, $\mathcal{C}$$\mathcal C$ may be regarded as a genuine categorification of $B$$B$ in the naive sense.

The relative category ${\mathsf{Cat}}_{/B}^{\mathrm{decat}}$$\mathsf{Cat}^{\operatorname{decat}}_{/B}$ is more flexible. It lets us study categories lying over $B$$B$, even when their naive decategorification only lands in a subset or substructure of $B$$B$.

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3. The Basic Example: $\mathbb{N}$$\mathbb N$

The simplest motivating example is the pair

$$\mathsf{FinSet},{\mathsf{FinVect}}_k.$$

The category $\mathsf{FinSet}$$\mathsf{FinSet}$ of finite sets decategorifies to $\mathbb{N}$$\mathbb N$ by cardinality:

$$[X]\mapsto |X|.$$

Indeed, two finite sets are isomorphic exactly when they have the same cardinality, so

$$\mathrm{Decat}(\mathsf{FinSet})\cong \mathbb{N}.$$

Similarly, the category ${\mathsf{FinVect}}_k$$\mathsf{FinVect}_k$ of finite-dimensional vector spaces over a field $k$$k$ decategorifies to $\mathbb{N}$$\mathbb N$ by dimension:

$$[V]\mapsto {\dim }_{k}V.$$

Two finite-dimensional vector spaces over $k$$k$ are isomorphic exactly when they have the same dimension, so

$$\mathrm{Decat}({\mathsf{FinVect}}_k)\cong \mathbb{N}.$$

Thus $\mathsf{FinSet}$$\mathsf{FinSet}$ and ${\mathsf{FinVect}}_k$$\mathsf{FinVect}_k$ give two categorifications of the same base $\mathbb{N}$$\mathbb N$.

The free vector space functor

$$k[-]:\mathsf{FinSet}\to {\mathsf{FinVect}}_k$$

satisfies

$${\dim }_{k}k[X]=|X|.$$

Therefore it preserves the decategorified value. In the relative category, it is a morphism

$$(\mathsf{FinSet},|-|)\to ({\mathsf{FinVect}}_k,{\dim }_k)$$

inside

$${\mathsf{Cat}}_{/\mathbb{N}}^{\mathrm{decat}}.$$

So the free vector space functor is not merely a familiar construction. It can be regarded as a morphism between two categorifications of $\mathbb{N}$$\mathbb N$.

If one remembers addition and multiplication, the example becomes richer. The set $\mathbb{N}$$\mathbb N$ is a semiring. The category $\mathsf{FinSet}$$\mathsf{FinSet}$ categorifies this semiring structure using disjoint union and Cartesian product:

$$X+Y:=X\bigsqcup Y,XY:=X\times Y.$$

The category ${\mathsf{FinVect}}_k$$\mathsf{FinVect}_k$ categorifies the same semiring structure using direct sum and tensor product:

$$V+W:=V\oplus W,VW:=V{\otimes }_{k}W.$$

The free vector space functor is compatible with both operations:

$$k[X\bigsqcup Y]\cong k[X]\oplus k[Y],$$

and

$$k[X\times Y]\cong k[X]{\otimes }_{k}k[Y].$$

Thus $k[-]$$k[-]$ is not only a morphism between categorifications of the underlying set $\mathbb{N}$$\mathbb N$. It is also a semiring-level linearization morphism.

This example suggests the general principle of the paper:

A categorification may be studied relative to a decategorified base, and a functor between categorifications should be regarded as a morphism over that base.

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4. Functorial Base Transformation under Decategorification

We now formulate a general mechanism.

Let

$$T:\mathsf{Cat}\to \mathsf{Cat}$$

be an endofunctor on categories. One might ask whether $T$$T$ automatically induces a functor

$${\mathsf{Cat}}_{/B}^{\mathrm{decat}}\to {\mathsf{Cat}}_{/B^{\prime }}^{\mathrm{decat}}$$

for suitable bases $B$$B$ and $B^{\prime }$$B'$.

The answer is no. The functor $T$$T$ alone is not enough. To transform categories over bases, one must also control how $T$$T$ behaves after decategorification.

A convenient sufficient condition is the following. Suppose we are given:

1. an endofunctor

$$T:\mathsf{Cat}\to \mathsf{Cat};$$

2. an endofunctor

$$S:\mathsf{Set}\to \mathsf{Set};$$

3. a natural transformation

$$\alpha :\mathrm{Decat}\circ T\Rightarrow S\circ \mathrm{Decat}.$$

Thus, for every category $\mathcal{C}$$\mathcal C$, there is a natural map

$${\alpha }_{\mathcal{C}}:\mathrm{Decat}(T\mathcal{C})\to S(\mathrm{Decat}(\mathcal{C})).$$

This says that the decategorification of $T\mathcal{C}$$T\mathcal C$ can be measured functorially in terms of the decategorification of $\mathcal{C}$$\mathcal C$.

Now fix a map of bases

$$\gamma :S(B)\to B^{\prime }.$$

Given an object

$$(\mathcal{C},p)\in {\mathsf{Cat}}_{/B}^{\mathrm{decat}},$$

where

$$p:\mathrm{Decat}(\mathcal{C})\to B,$$

define

$$p_T:\mathrm{Decat}(T\mathcal{C})\to B^{\prime }$$

by the composite

$$\mathrm{Decat}(T\mathcal{C})\stackrel{{\alpha }_{\mathcal{C}}}{\to }S(\mathrm{Decat}(\mathcal{C}))\stackrel{S(p)}{\to }S(B)\stackrel{\gamma }{\to }{B}^{\prime }.$$

In other words,

$$p_T=\gamma \circ S(p)\circ {\alpha }_{\mathcal{C}}.$$

Then $T$$T$ sends $(\mathcal{C},p)$$(\mathcal C,p)$ to

$$(T\mathcal{C},p_T)\in {\mathsf{Cat}}_{/B^{\prime }}^{\mathrm{decat}}.$$

This construction is functorial on morphisms. Suppose

$$F:(\mathcal{C},p)\to (\mathcal{D},q)$$

is a morphism in ${\mathsf{Cat}}_{/B}^{\mathrm{decat}}$$\mathsf{Cat}^{\operatorname{decat}}_{/B}$. Thus

$$q\circ \mathrm{Decat}(F)=p.$$

We claim that

$$T(F):T\mathcal{C}\to T\mathcal{D}$$

is a morphism

$$(T\mathcal{C},p_T)\to (T\mathcal{D},q_T)$$

over $B^{\prime }$$B'$.

Indeed, by naturality of $\alpha$$\alpha$, the square

$$\begin{array}{ccc}\mathrm{Decat}(T\mathcal{C})& \stackrel{\mathrm{Decat}(TF)}{\to }& \mathrm{Decat}(T\mathcal{D})\\ \downarrow {\alpha }_{\mathcal{C}}& & \downarrow {\alpha }_{\mathcal{D}}\\ S(\mathrm{Decat}(\mathcal{C}))& \stackrel{S(\mathrm{Decat}(F))}{\to }& S(\mathrm{Decat}(\mathcal{D}))\end{array}$$

commutes. Therefore

$$q_T\circ \mathrm{Decat}(TF)=\gamma \circ S(q)\circ {\alpha }_{\mathcal{D}}\circ \mathrm{Decat}(TF).$$

Using naturality, this becomes

$$\gamma \circ S(q)\circ S(\mathrm{Decat}(F))\circ {\alpha }_{\mathcal{C}}.$$

Since

$$q\circ \mathrm{Decat}(F)=p,$$

we get

$$S(q)\circ S(\mathrm{Decat}(F))=S(p).$$

Hence

$$q_T\circ \mathrm{Decat}(TF)=\gamma \circ S(p)\circ {\alpha }_{\mathcal{C}}=p_T.$$

Thus $TF$$TF$ preserves the transformed decategorified value.

We have proved the following proposition.

Proposition 4.1. Let $T:\mathsf{Cat}\to \mathsf{Cat}$$T:\mathsf{Cat}\to\mathsf{Cat}$ and $S:\mathsf{Set}\to \mathsf{Set}$$S:\mathsf{Set}\to\mathsf{Set}$ be functors. Suppose there is a natural transformation

$$\alpha :\mathrm{Decat}\circ T\Rightarrow S\circ \mathrm{Decat}.$$

For every map of bases

$$\gamma :S(B)\to B^{\prime },$$

there is an induced functor

$$T_{\gamma }:{\mathsf{Cat}}_{/B}^{\mathrm{decat}}\to {\mathsf{Cat}}_{/B^{\prime }}^{\mathrm{decat}}$$

defined by

$$(\mathcal{C},p)\mapsto (T\mathcal{C},\gamma \circ S(p)\circ {\alpha }_{\mathcal{C}})$$

and

$$F\mapsto T(F).$$

This proposition explains precisely in what sense an endofunctor on categories can induce a base transformation under decategorification. The essential data is not just $T$$T$, but the compatibility

$$\mathrm{Decat}\circ T\Rightarrow S\circ \mathrm{Decat}.$$

Applying $T$$T$ before decategorifying must be controlled by applying $S$$S$ after decategorifying.

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5. The Finite-Support $M$$M$-Grading Functor

We now apply the previous proposition to the main construction of the paper.

Let $M$$M$ be a monoid. Let $D(M)$$D(M)$ denote the discrete monoidal category associated to $M$$M$. Its objects are the elements of $M$$M$, its only morphisms are identities, and its tensor product is induced by multiplication in $M$$M$:

$$m\otimes n=mn.$$

The unit object is the identity element $e\in M$$e\in M$.

If $M$$M$ is commutative, then $D(M)$$D(M)$ is symmetric monoidal. If $M$$M$ is not commutative, then $D(M)$$D(M)$ is monoidal but not symmetric in general.

Let $\mathcal{C}$$\mathcal C$ be a category equipped with a chosen zero-like object $0_{\mathcal{C}}$$0_{\mathcal C}$. This could be the empty set in $\mathsf{FinSet}$$\mathsf{FinSet}$, or the zero vector space in ${\mathsf{FinVect}}_k$$\mathsf{FinVect}_k$.

Define

$$T_M(\mathcal{C})={\mathrm{Fun}}_{\mathrm{fs}}(D(M),\mathcal{C}),$$

the category of finite-support $M$$M$-indexed families of objects of $\mathcal{C}$$\mathcal C$.

Since $D(M)$$D(M)$ is discrete, an object of $T_M(\mathcal{C})$$T_M(\mathcal C)$ is simply a family

$$X=(X_m{)}_{m\in M}.$$

It has finite support if

$$X_m\ncong 0_{\mathcal{C}}$$

for only finitely many $m\in M$$m\in M$.

A morphism

$$X\to Y$$

is a family of morphisms

$$(f_m:X_m\to Y_m{)}_{m\in M}.$$

To make this construction functorial, we work with categories equipped with chosen zero-like objects and functors preserving them. If

$$F:\mathcal{C}\to \mathcal{D}$$

preserves the chosen zero-like objects, then $F$$F$ sends finite-support families to finite-support families. Hence it induces a functor

$$T_M(F):T_M(\mathcal{C})\to T_M(\mathcal{D})$$

by pointwise application:

$$(X_m{)}_{m\in M}\mapsto (F(X_m){)}_{m\in M}.$$

Thus

$$\mathcal{C}\mapsto {\mathrm{Fun}}_{\mathrm{fs}}(D(M),\mathcal{C})$$

is functorial in $\mathcal{C}$$\mathcal C$, on the appropriate category of categories with chosen zero-like objects.

At the set level, the corresponding construction is the finite-support function functor. If $A$$A$ is a pointed set with basepoint $a_0$$a_0$, define

$$S_M(A)=\{f:M\to A\mid f(m)\ne a_0\text{ for only finitely many }m\}.$$

For a category $\mathcal{C}$$\mathcal C$ with chosen zero-like object $0_{\mathcal{C}}$$0_{\mathcal C}$, the set

$$\mathrm{Decat}(\mathcal{C})$$

has a distinguished element $[0_{\mathcal{C}}]$$[0_{\mathcal C}]$. There is a natural map

$${\alpha }_{\mathcal{C}}:\mathrm{Decat}(T_M\mathcal{C})\to S_M(\mathrm{Decat}(\mathcal{C}))$$

defined by

$$[(X_m{)}_{m\in M}]\mapsto (m\mapsto [X_m]).$$

This is well-defined because isomorphic $M$$M$-graded objects have pointwise isomorphic components.

It is natural in $\mathcal{C}$$\mathcal C$. If $F:\mathcal{C}\to \mathcal{D}$$F:\mathcal C\to\mathcal D$ preserves the chosen zero-like objects, then

$${\alpha }_{\mathcal{D}}([(F(X_m){)}_{m\in M}])=(m\mapsto [F(X_m)]),$$

while

$$S_M(\mathrm{Decat}(F))({\alpha }_{\mathcal{C}}([(X_m)]))=S_M(\mathrm{Decat}(F))(m\mapsto [X_m])=(m\mapsto [F(X_m)]).$$

Therefore

$$\alpha :\mathrm{Decat}\circ T_M\Rightarrow S_M\circ \mathrm{Decat}$$

is a natural transformation.

Now take a category over $\mathbb{N}$$\mathbb N$,

$$(\mathcal{C},p)\in {\mathsf{Cat}}_{/\mathbb{N}}^{\mathrm{decat}},$$

where

$$p:\mathrm{Decat}(\mathcal{C})\to \mathbb{N}.$$

Assume

$$p([0_{\mathcal{C}}])=0.$$

Define the base transformation

$$\gamma :S_M(\mathbb{N})\to \mathbb{N}[M]$$

by

$$\gamma (f)=\sum _{m\in M}f(m)m.$$

Since $f$$f$ has finite support, this is a finite sum.

By Proposition 4.1, we obtain a functor

$$(-{)}^{(M)}:{\mathsf{Cat}}_{/\mathbb{N}}^{\mathrm{decat},0}\to {\mathsf{Cat}}_{/\mathbb{N}[M]}^{\mathrm{decat}},$$

where the source denotes the category of categories over $\mathbb{N}$$\mathbb N$ equipped with a chosen zero-like object, and whose morphisms preserve it.

Explicitly,

$$(\mathcal{C},p)\mapsto ({\mathcal{C}}^{(M)},p^{(M)}),$$

where

$${\mathcal{C}}^{(M)}={\mathrm{Fun}}_{\mathrm{fs}}(D(M),\mathcal{C})$$

and

$$p^{(M)}:\mathrm{Decat}({\mathcal{C}}^{(M)})\to \mathbb{N}[M]$$

is given by

$$p^{(M)}([(X_m{)}_{m\in M}])=\sum _{m\in M}p([X_m])m.$$

On morphisms,

$$F\mapsto F^{(M)}={\mathrm{Fun}}_{\mathrm{fs}}(D(M),F).$$

Thus forming finite-support $M$$M$-graded objects is not merely a way to build examples. It is a functorial operation

$$\text{categorifications over }\mathbb{N}⟶\text{categorifications over }\mathbb{N}[M].$$

This is the structural core of the monoid-ring construction.

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6. Day Convolution and the Monoid Semiring $\mathbb{N}[M]$$\mathbb N[M]$

We now add the monoidal structure.

Consider

$${\mathsf{Fin}}^{(M)}={\mathrm{Fun}}_{\mathrm{fs}}(D(M),\mathsf{FinSet}).$$

Objects of ${\mathsf{Fin}}^{(M)}$$\mathsf{Fin}^{(M)}$ are finite-support $M$$M$-graded finite sets

$$A=(A_m{)}_{m\in M}.$$

The monoidal structure on $D(M)$$D(M)$ induces, by Day convolution, a monoidal structure on ${\mathsf{Fin}}^{(M)}$$\mathsf{Fin}^{(M)}$. Since $D(M)$$D(M)$ is discrete, the formula is concrete:

$$(A\ast B{)}_r=\coprod _{mn=r}{A}_m\times B_n.$$

Equivalently, since $A$$A$ and $B$$B$ have finite support, the coproduct is taken over the finite set of pairs

$$(m,n)\in \mathrm{supp}(A)\times \mathrm{supp}(B)$$

such that $mn=r$$mn=r$.

The unit object is the graded finite set ${\delta }_e$$\delta_e$ defined by

$$({\delta }_e{)}_e=\{\ast \},({\delta }_e{)}_m=\varnothing (m\ne e).$$

Disjoint union gives addition, while Day convolution gives multiplication. The naive decategorification map is

$$[A]\mapsto \sum _{m\in M}|A_m|m.$$

This lands in the monoid semiring

$$\mathbb{N}[M].$$

Moreover,

$$[A\bigsqcup B]=[A]+[B],$$

and

$$[A\ast B]=[A]\cdot [B].$$

Indeed,

$$|(A\ast B{)}_r|=\sum _{mn=r}|A_m||B_n|,$$

which is exactly the coefficient of $r$$r$ in the product

$$(\sum _m|A_m|m)(\sum _n|B_n|n).$$

If ${\delta }_m$$\delta_m$ denotes the object concentrated at $m$$m$ with value a one-point set, then

$${\delta }_m\ast {\delta }_n\cong {\delta }_{mn}.$$

Thus

$$[{\delta }_m]\cdot [{\delta }_n]=[{\delta }_{mn}].$$

This is precisely the multiplication law in $\mathbb{N}[M]$$\mathbb N[M]$.

Therefore

$${\mathsf{Fin}}^{(M)}$$

categorifies the monoid semiring $\mathbb{N}[M]$$\mathbb N[M]$, with disjoint union categorifying addition and Day convolution categorifying multiplication.

If $M$$M$ is commutative, then the monoidal structure is symmetric. If $M$$M$ is noncommutative, then the convolution is generally monoidal but not symmetric, reflecting the noncommutativity of $\mathbb{N}[M]$$\mathbb N[M]$.

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7. $M$$M$-Graded Vector Spaces and the Monoid Ring

Now consider

$${\mathsf{Vect}}_k^{(M)}={\mathrm{Fun}}_{\mathrm{fs}}(D(M),{\mathsf{FinVect}}_k).$$

An object is a finite-support $M$$M$-graded finite-dimensional vector space

$$V=(V_m{)}_{m\in M}.$$

Day convolution gives

$$(V\ast W{)}_r=\underset{mn=r}{⨁}{V}_m{\otimes }_k{W}_n.$$

The unit object is concentrated at the identity element $e$$e$, with value $k$$k$ at $e$$e$ and $0$$0$ elsewhere.

The naive decategorification map is

$$[V]\mapsto \sum _{m\in M}{\dim }_k(V_m)m.$$

Its image lies in

$$\mathbb{N}[M]\subset \mathbb{Z}[M].$$

Thus, under naive decategorification, ${\mathsf{Vect}}_k^{(M)}$$\mathsf{Vect}_k^{(M)}$ gives $\mathbb{N}[M]$$\mathbb N[M]$, not the whole monoid ring $\mathbb{Z}[M]$$\mathbb Z[M]$. This is expected: dimensions are nonnegative integers.

To obtain $\mathbb{Z}[M]$$\mathbb Z[M]$, one passes to the Grothendieck group. The additive structure is direct sum:

$$(V\oplus W{)}_m=V_m\oplus W_m.$$

Since finite-dimensional vector spaces are semisimple, and all constructions here are pointwise, the Grothendieck group is obtained by group-completing the commutative monoid of isomorphism classes under direct sum:

$$K_0({\mathsf{Vect}}_k^{(M)})\cong \mathbb{Z}[M].$$

Explicitly, the class of $V=(V_m{)}_{m\in M}$$V=(V_m)_{m\in M}$ is sent to

$$[V]\mapsto \sum _{m\in M}{\dim }_k(V_m)m.$$

Negative coefficients in $\mathbb{Z}[M]$$\mathbb Z[M]$ do not come from actual objects. They arise from formal differences

$$[V]-[W]$$

in the Grothendieck group.

Day convolution induces multiplication on $K_0$$K_0$:

$$[V]\cdot [W]=[V\ast W].$$

For the object ${\delta }_m$$\delta_m$ concentrated at $m$$m$ with value $k$$k$, one has

$${\delta }_m\ast {\delta }_n\cong {\delta }_{mn}.$$

Hence

$$[{\delta }_m]\cdot [{\delta }_n]=[{\delta }_{mn}].$$

Therefore,

$$K_0({\mathsf{Vect}}_k^{(M)},\oplus ,\ast )\cong \mathbb{Z}[M]$$

as rings.

The process can be summarized as

$$({\mathsf{Vect}}_k^{(M)}{)}^{\simeq }\stackrel{{\pi }_0}{\to }\mathbb{N}[M]\stackrel{\mathrm{Gr}}{\to }\mathbb{Z}[M].$$

Here ${\pi }_0$$\pi_0$ records the positive part of the categorified structure, while $K_0$$K_0$ performs the group completion that produces the ring.

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8. Linearization and the Lifted Free Vector Space Functor

The free vector space functor

$$k[-]:\mathsf{FinSet}\to {\mathsf{FinVect}}_k$$

is a morphism over $\mathbb{N}$$\mathbb N$, since

$${\dim }_{k}k[X]=|X|.$$

Applying the finite-support $M$$M$-grading functor gives

$$k[-{]}^{(M)}:={\mathrm{Fun}}_{\mathrm{fs}}(D(M),k[-]):{\mathsf{Fin}}^{(M)}\to {\mathsf{Vect}}_k^{(M)}.$$

Explicitly,

$$k[-{]}^{(M)}((A_m{)}_{m\in M})=(k[A_m]{)}_{m\in M}.$$

This is the pointwise free vector space functor.

It preserves the decategorified value over $\mathbb{N}[M]$$\mathbb N[M]$, because

$${\dim }_{k}k[A_m]=|A_m|$$

for each $m\in M$$m\in M$. Therefore

$$\sum _{m\in M}|A_m|m\mapsto \sum _{m\in M}{\dim }_k(k[A_m])m=\sum _{m\in M}|A_m|m.$$

Thus $k[-{]}^{(M)}$$k[-]^{(M)}$ induces the identity map on $\mathbb{N}[M]$$\mathbb N[M]$ after naive decategorification.

Moreover, $k[-{]}^{(M)}$$k[-]^{(M)}$ is compatible with Day convolution. Indeed,

$$k[(A\ast B{)}_r]=k[\coprod _{mn=r}{A}_m\times B_n].$$

Since the free vector space functor sends finite coproducts of sets to direct sums of vector spaces and finite Cartesian products to tensor products, we have natural isomorphisms

$$k[\coprod _{mn=r}{A}_m\times B_n]\cong \underset{mn=r}{⨁}k[A_m\times B_n]\cong \underset{mn=r}{⨁}k[A_m]{\otimes }_{k}k[B_n].$$

The right-hand side is precisely

$$(k[A]\ast k[B]{)}_r.$$

Therefore

$$k[A\ast B]\cong k[A]\ast k[B].$$

The unit is also preserved:

$$k[{\delta }_e]\cong {\delta }_e.$$

Hence

$$k[-{]}^{(M)}:{\mathsf{Fin}}^{(M)}\to {\mathsf{Vect}}_k^{(M)}$$

is a strong monoidal functor for Day convolution.

This gives the lifted diagram

$$\mathsf{FinSet}⟶{\mathsf{FinVect}}_k$$

over $\mathbb{N}$$\mathbb N$, transformed into

$${\mathrm{Fun}}_{\mathrm{fs}}(D(M),\mathsf{FinSet})⟶{\mathrm{Fun}}_{\mathrm{fs}}(D(M),{\mathsf{FinVect}}_k)$$

over $\mathbb{N}[M]$$\mathbb N[M]$.

The passage from $\mathbb{N}[M]$$\mathbb N[M]$ to $\mathbb{Z}[M]$$\mathbb Z[M]$ then occurs on the vector-space side by applying $K_0$$K_0$.

Thus the full picture has two stages:

$$\text{categorifications over }\mathbb{N}\stackrel{(-{)}^{(M)}}{\to }\text{categorifications over }\mathbb{N}[M]\stackrel{K_0}{\to }\text{ring-level decategorification over }\mathbb{Z}[M].$$

The first stage is functorial at the level of categories over a base. The second stage is Grothendieck group completion.

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9. The Relative Viewpoint Revisited

The relative viewpoint is useful because it avoids an overly rigid notion of categorification.

For example, if we take

$$B=\mathbb{Z}[M],$$

then ${\mathsf{Vect}}_k^{(M)}$$\mathsf{Vect}_k^{(M)}$ has a natural map

$$\mathrm{Decat}({\mathsf{Vect}}_k^{(M)})\to \mathbb{Z}[M],$$

given by

$$[V]\mapsto \sum _{m\in M}{\dim }_k(V_m)m.$$

The image is only

$$\mathbb{N}[M]\subset \mathbb{Z}[M].$$

Nevertheless, this still defines an object of

$${\mathsf{Cat}}_{/\mathbb{Z}[M]}^{\mathrm{decat}}.$$

Thus the comma category

$$(\mathrm{Decat}\downarrow B)$$

allows us to study categories whose naive decategorification maps into $B$$B$, even when it does not exhaust $B$$B$.

If we require the stronger condition

$$\mathrm{Decat}(\mathcal{C})\cong B,$$

then ${\mathsf{Vect}}_k^{(M)}$$\mathsf{Vect}_k^{(M)}$ would not be a genuine naive categorification of $\mathbb{Z}[M]$$\mathbb Z[M]$. It is instead a category over $\mathbb{Z}[M]$$\mathbb Z[M]$ whose naive image is the positive part $\mathbb{N}[M]$$\mathbb N[M]$. To recover the whole ring, one must pass from ${\pi }_0$$\pi_0$ to $K_0$$K_0$.

This distinction is important:

$${\pi }_0\text{remembers object classes and gives positive coefficients;}$$

$$K_0\text{group-completes the additive structure and gives negative coefficients.}$$

Thus the relative framework separates two operations that are often conflated:

1. assigning decategorified values to objects;

2. completing the additive structure to produce a group or ring.

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10. Summary

Let $M$$M$ be a monoid.

The category

$${\mathsf{Fin}}^{(M)}={\mathrm{Fun}}_{\mathrm{fs}}(D(M),\mathsf{FinSet})$$

of finite-support $M$$M$-graded finite sets categorifies the monoid semiring $\mathbb{N}[M]$$\mathbb N[M]$. Addition is given by disjoint union, and multiplication is given by Day convolution.

The category

$${\mathsf{Vect}}_k^{(M)}={\mathrm{Fun}}_{\mathrm{fs}}(D(M),{\mathsf{FinVect}}_k)$$

of finite-support $M$$M$-graded finite-dimensional vector spaces has naive decategorification $\mathbb{N}[M]$$\mathbb N[M]$. After passing to the Grothendieck group, it gives the monoid ring:

$$K_0({\mathsf{Vect}}_k^{(M)})\cong \mathbb{Z}[M].$$

The free vector space functor

$$k[-]:\mathsf{FinSet}\to {\mathsf{FinVect}}_k$$

is a morphism between categorifications over $\mathbb{N}$$\mathbb N$. By functoriality of the finite-support $M$$M$-grading construction, it lifts to

$$k[-{]}^{(M)}:{\mathrm{Fun}}_{\mathrm{fs}}(D(M),\mathsf{FinSet})\to {\mathrm{Fun}}_{\mathrm{fs}}(D(M),{\mathsf{FinVect}}_k).$$

This lifted functor is strong monoidal for Day convolution and induces the identity on $\mathbb{N}[M]$$\mathbb N[M]$ after naive decategorification.

The structural mechanism is the functor

$$(-{)}^{(M)}={\mathrm{Fun}}_{\mathrm{fs}}(D(M),-),$$

which sends categorifications over $\mathbb{N}$$\mathbb N$ to categorifications over $\mathbb{N}[M]$$\mathbb N[M]$. This mechanism is an instance of a more general principle: an endofunctor $T$$T$ on categories induces a transformation between categorifications over bases only when its decategorification is controlled by compatible data

$$\mathrm{Decat}\circ T\Rightarrow S\circ \mathrm{Decat}.$$

In the present case, $T$$T$ is finite-support $M$$M$-grading, $S$$S$ is the finite-support function construction, and the base transformation sends a finite-support function $M\to \mathbb{N}$$M\to\mathbb N$ to the corresponding element of $\mathbb{N}[M]$$\mathbb N[M]$.

Thus the monoid ring construction is not an isolated computation. It is part of a functorial theory of how categorifications transform under controlled operations on categories.