A Preadditive Syntax for Chain Complexes: From Sketch Models to Homology Functors

The Preadditive Sketch of Chain Complexes

The purpose of this note is to describe chain complexes from a syntactic point of view.
The basic idea is simple: a chain complex is a model of a small preadditive sketch.
This separates two layers:

1. The syntactic layer, where the shape of a chain complex is freely generated.

2. The semantic layer, where a model interprets this syntax inside a preadditive or abelian category.

From this perspective, the category of chain complexes in an abelian category is automatically abelian, because it is a category of additive functors into an abelian category.
Moreover, the differential itself can be seen as a universal natural transformation in the syntactic category. This universal differential then induces the usual differential natural transformation on the category of models. Cycles, boundaries, and homology can then be constructed inside an additive endofunctor category by taking kernels, images, and cokernels.

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Universe Convention

We fix a Grothendieck universe $\mathcal{U}$$\mathcal{U}$.
A set, category, functor, or natural transformation is called small if it is small relative to $\mathcal{U}$$\mathcal{U}$, unless otherwise stated.

When forming functor categories whose source is already large relative to $\mathcal{U}$$\mathcal{U}$, we pass to a sufficiently larger Grothendieck universe $\mathcal{V}$$\mathcal{V}$ with $\mathcal{U}\in \mathcal{V}$$\mathcal{U}\in\mathcal{V}$.
Thus, all functor categories in this note are formed in a universe large enough for the relevant source category to be small.

This convention is only a size convention. It does not change the mathematical constructions, which are all computed pointwise.

For example, if $R$$R$ is a $\mathcal{U}$$\mathcal{U}$-small ring, then $R\text{-}{\mathsf{Mod}}_{\mathcal{U}}$$R\text{-}\mathsf{Mod}_{\mathcal{U}}$ denotes the category of $\mathcal{U}$$\mathcal{U}$-small left $R$$R$-modules. This category is usually not $\mathcal{U}$$\mathcal{U}$-small, but it is small in a sufficiently larger universe $\mathcal{V}$$\mathcal{V}$.
Similarly, ${\mathsf{Ch}}_{\mathcal{U}}(R)$$\mathsf{Ch}_{\mathcal{U}}(R)$ denotes the category of chain complexes of $\mathcal{U}$$\mathcal{U}$-small left $R$$R$-modules. When we form its additive endofunctor category, we do so inside a sufficiently large universe.

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The Syntactic Category

Define a preadditive category ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ as follows.

Its objects are the integers:

$$\mathrm{Ob}({\mathcal{C}}_{\mathrm{ch}})=\mathbb{Z}.$$

For every integer $n$$n$, we freely add a generating morphism:

$$d_n:n⟶n-1.$$

The only essential relation is:

$$d_{n-1}\circ d_n=0.$$

Because ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ is preadditive, every Hom-set is an abelian group. Thus morphisms can be added, subtracted, and multiplied by integers. In particular, zero morphisms exist.
This is why the relation $d_{n-1}\circ d_n=0$$d_{n-1}\circ d_n=0$ is meaningful.

Note that ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ is not required to have a zero object, nor is it required to have finite biproducts.
For the syntax of a chain complex, we only need zero morphisms, not a zero object. Zero morphisms already exist in any preadditive category because every Hom-set is an abelian group.

Therefore, the minimal syntactic structure for chain complexes is preadditive, not additive.

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The Hom Groups

The Hom groups in ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ are generated by identities and the arrows $d_n$$d_n$.

Informally, one has:

$${\mathrm{Hom}}_{{\mathcal{C}}_{\mathrm{ch}}}(n,n)\cong \mathbb{Z}\cdot {\mathrm{id}}_n,$$

$${\mathrm{Hom}}_{{\mathcal{C}}_{\mathrm{ch}}}(n,n-1)\cong \mathbb{Z}\cdot d_n.$$

Since the relation $d_{n-1}{d}_n=0$$d_{n-1}d_n=0$ kills all paths of length two, there are no nonzero composites of two consecutive differentials.

Thus, for $m<n-1$$m<n-1$, one has:

$${\mathrm{Hom}}_{{\mathcal{C}}_{\mathrm{ch}}}(n,m)=0.$$

The category ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ is therefore the free preadditive syntactic category generated by a chain-shaped sequence of arrows whose consecutive composites vanish.

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Models of the Sketch

Let $\mathcal{A}$$\mathcal{A}$ be a preadditive category.
A model of the sketch ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ in $\mathcal{A}$$\mathcal{A}$ is an additive functor:

$$F:{\mathcal{C}}_{\mathrm{ch}}⟶\mathcal{A}.$$

Here, additive means that for every pair of objects $x,y$$x,y$ in ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$, the induced map

$${\mathrm{Hom}}_{{\mathcal{C}}_{\mathrm{ch}}}(x,y)⟶{\mathrm{Hom}}_{\mathcal{A}}(F(x),F(y))$$

is a homomorphism of abelian groups.

The functor $F$$F$ assigns to each integer $n$$n$ an object:

$$A_n=F(n).$$

It also assigns to each generating morphism $d_n:n\to n-1$$d_n:n\to n-1$ a morphism:

$${\partial }_n=F(d_n):A_n⟶A_{n-1}.$$

Since $F$$F$ preserves composition and zero morphisms, the syntactic relation $d_{n-1}\circ d_n=0$$d_{n-1}\circ d_n=0$ becomes:

$${\partial }_{n-1}\circ {\partial }_n=0.$$

Thus, a model of ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ in $\mathcal{A}$$\mathcal{A}$ is precisely a chain complex in $\mathcal{A}$$\mathcal{A}$.

Therefore:

$$\mathsf{Ch}(\mathcal{A})\simeq \mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A}).$$

Here $\mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A})$$\mathsf{Add}(\mathcal{C}_{\mathrm{ch}},\mathcal{A})$ denotes the category of additive functors from ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ to $\mathcal{A}$$\mathcal{A}$, with natural transformations as morphisms.

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Chain Maps as Natural Transformations

Let

$$F,G:{\mathcal{C}}_{\mathrm{ch}}⟶\mathcal{A}$$

be two additive functors. They correspond to two chain complexes:

$$A_n=F(n),B_n=G(n).$$

A natural transformation

$$\alpha :F⟹G$$

is a family of morphisms:

$${\alpha }_n:A_n⟶B_n.$$

Naturality with respect to the generating morphism $d_n:n\to n-1$$d_n:n\to n-1$ says:

$$G(d_n)\circ {\alpha }_n={\alpha }_{n-1}\circ F(d_n).$$

In chain complex notation, this becomes:

$${\partial }_n^B\circ {\alpha }_n={\alpha }_{n-1}\circ {\partial }_n^A.$$

This is exactly the chain map condition.
Therefore, chain maps are not an additional notion; they are precisely natural transformations between models of the preadditive sketch.

Hence the category of chain complexes in $\mathcal{A}$$\mathcal{A}$ is the additive functor category:

$$\mathsf{Ch}(\mathcal{A})\simeq \mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A}).$$

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Why the Category of Chain Complexes Is Abelian

Now assume that $\mathcal{A}$$\mathcal{A}$ is an abelian category.

Since ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ is a small preadditive category, the additive functor category

$$\mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A})$$

is an abelian category. Kernels, cokernels, images, and coimages are computed pointwise.

For example, let

$$\alpha :F⟹G$$

be a natural transformation between additive functors. Then its kernel is the additive functor $\ker \alpha$$\ker\alpha$ defined by:

$$(\ker \alpha )(n)=\ker ({\alpha }_n:F(n)\to G(n)).$$

Similarly, its cokernel is defined by:

$$(\mathrm{coker}\alpha )(n)=\mathrm{coker}({\alpha }_n:F(n)\to G(n)).$$

The image is defined pointwise by:

$$(\mathrm{im}\alpha )(n)=\mathrm{im}({\alpha }_n:F(n)\to G(n)).$$

Since $\mathcal{A}$$\mathcal{A}$ is abelian, the canonical map from coimage to image is an isomorphism at each degree. Hence it is an isomorphism in the functor category.

Therefore:

$$\mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A})\text{is abelian}.$$

Using the identification

$$\mathsf{Ch}(\mathcal{A})\simeq \mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A}),$$

we obtain:

$$\mathsf{Ch}(\mathcal{A})\text{is an abelian category}.$$

This proof does not require manually checking that kernels and cokernels of chain maps are chain complexes. Instead, it follows from a general theorem about additive functor categories.

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The Case of $R$$R$-Modules

Let $R$$R$ be a $\mathcal{U}$$\mathcal{U}$-small ring, and let $R\text{-}{\mathsf{Mod}}_{\mathcal{U}}$$R\text{-}\mathsf{Mod}_{\mathcal{U}}$ be the category of $\mathcal{U}$$\mathcal{U}$-small left $R$$R$-modules. This is an abelian category, locally small relative to $\mathcal{U}$$\mathcal{U}$, and small in a sufficiently larger universe.

Therefore, the category of chain complexes of $R$$R$-modules is:

$${\mathsf{Ch}}_{\mathcal{U}}(R)=\mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},R\text{-}{\mathsf{Mod}}_{\mathcal{U}}).$$

It is an abelian category.

This recovers the usual category of chain complexes of $R$$R$-modules, with kernels, cokernels, images, and coimages computed degreewise.

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The Syntactic Shift Functor

The syntactic category ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ has a natural shift endofunctor.

Define

$$S:{\mathcal{C}}_{\mathrm{ch}}⟶{\mathcal{C}}_{\mathrm{ch}}$$

on objects by:

$$S(n)=n-1.$$

There are two common choices for the action on generating morphisms.

The unsigned shift:

$$S_{+}(d_n)=d_{n-1}.$$

The signed shift:

$$S_{-}(d_n)=-d_{n-1}.$$

The signed version corresponds to the usual shift convention for chain complexes.

In the signed case, if $F:{\mathcal{C}}_{\mathrm{ch}}\to \mathcal{A}$$F:\mathcal{C}_{\mathrm{ch}}\to\mathcal{A}$ is a model, then $F\circ S_{-}$$F\circ S_-$ is the shifted complex with:

$$(F\circ S_{-})(n)=F(n-1).$$

The differential is:

$$F(S_{-}(d_n))=F(-d_{n-1})=-F(d_{n-1}).$$

Thus, if $F$$F$ corresponds to a chain complex $C$$C$, then $F\circ S_{-}$$F\circ S_-$ corresponds to the shifted complex $C[1]$$C[1]$ with

$$C[1{]}_n=C_{n-1}$$

and differential

$$d_n^{C[1]}=-d_{n-1}^C.$$

This is the usual homological shift convention.

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The Universal Differential

Inside the syntactic category ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$, there is a universal natural transformation:

$$\eta :{\mathrm{Id}}_{{\mathcal{C}}_{\mathrm{ch}}}⟹S.$$

Its component at $n$$n$ is the generating morphism:

$${\eta }_n=d_n:n⟶n-1=S(n).$$

This natural transformation is the universal differential.

For the unsigned shift $S_{+}$$S_+$, naturality with respect to $d_n$$d_n$ gives

$$S_{+}(d_n)\circ {\eta }_n={\eta }_{n-1}\circ d_n,$$

i.e. $d_{n-1}\circ d_n=d_{n-1}\circ d_n$$d_{n-1}\circ d_n = d_{n-1}\circ d_n$, which holds trivially.

For the signed shift $S_{-}$$S_-$, naturality gives

$$S_{-}(d_n)\circ {\eta }_n={\eta }_{n-1}\circ d_n,$$

i.e. $(-d_{n-1})\circ d_n=d_{n-1}\circ d_n$$(-d_{n-1})\circ d_n = d_{n-1}\circ d_n$.
But the defining relation in ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ is $d_{n-1}\circ d_n=0$$d_{n-1}\circ d_n=0$, so both sides are zero and naturality still holds.

Thus, even for the usual signed shift, the family ${\eta }_n=d_n$$\eta_n=d_n$ defines a natural transformation:

$$\eta :{\mathrm{Id}}_{{\mathcal{C}}_{\mathrm{ch}}}⟹S_{-}.$$

This shows that the differential is not merely an operation inside each model; it already exists syntactically as a universal natural transformation.

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The Induced Shift on the Model Category

Let

$$\mathcal{M}=\mathsf{Ch}(\mathcal{A})=\mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A}).$$

The syntactic shift $S:{\mathcal{C}}_{\mathrm{ch}}\to {\mathcal{C}}_{\mathrm{ch}}$$S:\mathcal{C}_{\mathrm{ch}}\to\mathcal{C}_{\mathrm{ch}}$ induces a shift endofunctor on the model category by precomposition:

$$[1]:\mathcal{M}⟶\mathcal{M},F[1]=F\circ S.$$

If $S=S_{-}$$S=S_-$, this is the usual signed shift.

The universal differential $\eta :{\mathrm{Id}}_{{\mathcal{C}}_{\mathrm{ch}}}⟹S$$\eta:\operatorname{Id}_{\mathcal{C}_{\mathrm{ch}}}\Longrightarrow S$ induces a natural transformation on the model category:

$$\delta :{\mathrm{Id}}_{\mathcal{M}}⟹[1].$$

For each chain complex $F$$F$, the component ${\delta }_F:F\to F[1]$$\delta_F:F\to F[1]$ has degree $n$$n$ part:

$$({\delta }_F{)}_n=F(d_n):F(n)⟶F(n-1).$$

In ordinary chain complex notation, this is simply:

$$({\delta }_C{)}_n=d_n^C:C_n⟶C_{n-1}=C[1{]}_n.$$

Thus, the differential of every chain complex is the interpretation of the universal syntactic differential.

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The Additive Endofunctor Category

Now assume that $\mathcal{A}$$\mathcal{A}$ is an abelian category and let $\mathcal{M}=\mathsf{Ch}(\mathcal{A})$$\mathcal{M}=\mathsf{Ch}(\mathcal{A})$.

Working in a sufficiently large Grothendieck universe, the additive endofunctor category

$$\mathsf{Add}(\mathcal{M},\mathcal{M})$$

is an abelian category.

Its objects are additive functors $T:\mathcal{M}⟶\mathcal{M}$$T:\mathcal{M}\longrightarrow\mathcal{M}$, and its morphisms are natural transformations.
Kernels, cokernels, images, and coimages are computed pointwise.

For a natural transformation $\alpha :F⟹G$$\alpha:F\Longrightarrow G$ between additive endofunctors, its kernel is the endofunctor defined by:

$$(\ker \alpha )(C)=\ker ({\alpha }_C:F(C)\to G(C)).$$

Its image is defined by:

$$(\mathrm{im}\alpha )(C)=\mathrm{im}({\alpha }_C:F(C)\to G(C)).$$

Its cokernel is defined by:

$$(\mathrm{coker}\alpha )(C)=\mathrm{coker}({\alpha }_C:F(C)\to G(C)).$$

Since $\mathcal{M}$$\mathcal{M}$ is abelian, these pointwise constructions define additive endofunctors of $\mathcal{M}$$\mathcal{M}$.

Therefore $\mathsf{Add}(\mathcal{M},\mathcal{M})$$\mathsf{Add}(\mathcal{M},\mathcal{M})$ is abelian. This is the correct environment in which to define cycles, boundaries, and homology as functors.

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Cycles as a Kernel in the Endofunctor Category

We have the natural transformation:

$$\delta :{\mathrm{Id}}_{\mathcal{M}}⟹[1].$$

Define the cycle endofunctor $Z$$Z$ by:

$$Z=\ker (\delta ).$$

This kernel is taken in the abelian category $\mathsf{Add}(\mathcal{M},\mathcal{M})$$\mathsf{Add}(\mathcal{M},\mathcal{M})$. Therefore $Z$$Z$ is automatically an additive endofunctor:

$$Z:\mathcal{M}⟶\mathcal{M}.$$

For a chain complex $C$$C$, we have:

$$Z(C)=\ker ({\delta }_C:C\to C[1]).$$

Degreewise, this gives:

$$Z(C{)}_n=\ker (d_n^C:C_n\to C_{n-1}).$$

Thus $Z(C{)}_n$$Z(C)_n$ is the usual object of $n$$n$-cycles.

The important point is that $Z$$Z$ is not first defined degreewise and then checked to be functorial. Instead, it is defined as a kernel in an abelian endofunctor category, so its functoriality is automatic.

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Boundaries as an Image in the Endofunctor Category

To define boundaries, we use the shifted version of the differential.

There is a natural transformation:

$${\delta }^{-}:[-1]⟹{\mathrm{Id}}_{\mathcal{M}}.$$

For a chain complex $C$$C$, the component ${\delta }_C^{-}$$\delta^-_C$ has degree $n$$n$ part given, up to the standard shift sign, by:

$$d_{n+1}^C:C_{n+1}⟶C_n.$$

The sign does not affect the image.

Define the boundary endofunctor $B$$B$ by:

$$B=\mathrm{im}({\delta }^{-}).$$

This image is taken in the abelian category $\mathsf{Add}(\mathcal{M},\mathcal{M})$$\mathsf{Add}(\mathcal{M},\mathcal{M})$. Therefore $B$$B$ is automatically an additive endofunctor:

$$B:\mathcal{M}⟶\mathcal{M}.$$

For a chain complex $C$$C$, we have:

$$B(C)=\mathrm{im}({\delta }_C^{-}).$$

Degreewise, this gives:

$$B(C{)}_n=\mathrm{im}(d_{n+1}^C:C_{n+1}\to C_n).$$

Thus $B(C{)}_n$$B(C)_n$ is the usual object of $n$$n$-boundaries. Again, functoriality is automatic.

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Boundaries Lie in Cycles

In the additive endofunctor category, we have a composite:

$$[-1]\stackrel{{\delta }^{-}}{\to }{\mathrm{Id}}_{\mathcal{M}}\stackrel{\delta }{\to }[1],$$

and this composite is zero:

$$\delta \circ {\delta }^{-}=0.$$

Degreewise, this is exactly the relation $d_n^C\circ d_{n+1}^C=0$$d^C_n\circ d^C_{n+1}=0$.

Therefore the image of ${\delta }^{-}$$\delta^-$ factors through the kernel of $\delta$$\delta$. That is, there is a natural monomorphism:

$$B\hookrightarrow Z.$$

This is the functorial version of the familiar inclusion $B_n(C)\subseteq Z_n(C)$$B_n(C)\subseteq Z_n(C)$, but here it follows directly from the image-kernel factorization in the abelian endofunctor category rather than a degreewise proof.

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Homology as a Cokernel in the Endofunctor Category

Since we have a natural monomorphism $B\hookrightarrow Z$$B\hookrightarrow Z$, we define the homology endofunctor $H$$H$ by:

$$H=\mathrm{coker}(B\hookrightarrow Z).$$

This cokernel is taken in the abelian category $\mathsf{Add}(\mathcal{M},\mathcal{M})$$\mathsf{Add}(\mathcal{M},\mathcal{M})$. Therefore $H$$H$ is automatically an additive endofunctor:

$$H:\mathcal{M}⟶\mathcal{M}.$$

For a chain complex $C$$C$, we get:

$$H(C)=\mathrm{coker}(B(C)\hookrightarrow Z(C)).$$

Degreewise:

$$H(C{)}_n=Z(C{)}_n/B(C{)}_n.$$

Equivalently:

$$H(C{)}_n=\ker (d_n^C:C_n\to C_{n-1})/\mathrm{im}(d_{n+1}^C:C_{n+1}\to C_n).$$

Thus ordinary homology arises as a cokernel in the additive endofunctor category.

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The Whole Construction

The entire structure is generated by the syntactic chain:

$$[-1]\stackrel{{\delta }^{-}}{\to }{\mathrm{Id}}_{\mathcal{M}}\stackrel{\delta }{\to }[1].$$

The relation $\delta \circ {\delta }^{-}=0$$\delta\circ\delta^-=0$ is the functorial form of the chain complex axiom $d^2=0$$d^2=0$.

Then:

• $B=\mathrm{im}({\delta }^{-})$$B=\operatorname{im}(\delta^-)$,

• $Z=\ker (\delta )$$Z=\ker(\delta)$,

• $H=\mathrm{coker}(B\hookrightarrow Z)$$H=\operatorname{coker}(B\hookrightarrow Z)$.

All of these constructions take place inside the abelian category $\mathsf{Add}(\mathcal{M},\mathcal{M})$$\mathsf{Add}(\mathcal{M},\mathcal{M})$.
Cycles, boundaries, and homology are not merely degreewise constructions; they are universal constructions in an endofunctor category.

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Conceptual Summary

The preadditive sketch of chain complexes consists of:

1. Objects indexed by integers $n\in \mathbb{Z}$$n\in\mathbb{Z}$.

2. Generating arrows $d_n:n\to n-1$$d_n:n\to n-1$.

3. Additive structure on Hom groups.

4. The relation $d_{n-1}{d}_n=0$$d_{n-1}d_n=0$.

Its free preadditive syntactic category is ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$.

For any preadditive category $\mathcal{A}$$\mathcal{A}$, a chain complex in $\mathcal{A}$$\mathcal{A}$ is an additive functor:

$$F:{\mathcal{C}}_{\mathrm{ch}}⟶\mathcal{A}.$$

For any abelian category $\mathcal{A}$$\mathcal{A}$, the category of chain complexes is:

$$\mathsf{Ch}(\mathcal{A})\simeq \mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A}).$$

Since ${\mathcal{C}}_{\mathrm{ch}}$$\mathcal{C}_{\mathrm{ch}}$ is small and $\mathcal{A}$$\mathcal{A}$ is abelian, this functor category is abelian.

The syntactic category also contains a universal differential:

$$\eta :{\mathrm{Id}}_{{\mathcal{C}}_{\mathrm{ch}}}⟹S,$$

which induces the differential natural transformation on the model category:

$$\delta :{\mathrm{Id}}_{\mathsf{Ch}(\mathcal{A})}⟹[1].$$

Working in a sufficiently large Grothendieck universe, the additive endofunctor category $\mathsf{Add}(\mathsf{Ch}(\mathcal{A}),\mathsf{Ch}(\mathcal{A}))$$\mathsf{Add}(\mathsf{Ch}(\mathcal{A}),\mathsf{Ch}(\mathcal{A}))$ is abelian.

Inside it, we define:

• $Z=\ker (\delta )$$Z=\ker(\delta)$,

• $B=\mathrm{im}({\delta }^{-})$$B=\operatorname{im}(\delta^-)$,

• $H=\mathrm{coker}(B\hookrightarrow Z)$$H=\operatorname{coker}(B\hookrightarrow Z)$.

This gives a fully functorial construction of cycles, boundaries, and homology.

The chain complex axiom $d^2=0$$d^2=0$ is already encoded in a small preadditive syntactic category. Once this syntax is interpreted in an abelian category, the usual abelian structure of chain complexes, the shift functor, the differential, cycles, boundaries, and homology all arise from functorial universal constructions.

Collaboration Report (With GPT 5.5 Thinking): The Preadditive Sketch of Chain Complexes

This report records the collaborative development of the note The Preadditive Sketch of Chain Complexes. The project aimed to reinterpret the category of chain complexes through a syntactic and functorial framework: a chain complex is treated as a model of a small preadditive sketch, and cycles, boundaries, and homology are then reconstructed as universal constructions in an additive endofunctor category.

The final text presents a unified categorical account of chain complexes. It begins with a freely generated preadditive syntactic category, interprets its models as additive functors into a preadditive or abelian category, derives the abelian structure of the chain complex category from the general theory of additive functor categories, and then uses the induced shift functor and universal differential to construct cycles, boundaries, and homology functorially.

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Marco's Contributions

Marco supplied the central mathematical idea of the project: that the syntax of a chain complex should be described not by an additive category with zero object and biproducts, but by a smaller preadditive sketch.

The key insight was that the relation $d^2=0$$d^2=0$ does not require a zero object. It only requires zero morphisms, and zero morphisms already exist in any preadditive category because each Hom is an abelian group. This distinction between zero morphisms and zero objects became one of the conceptual foundations of the note.

Marco also identified the correct syntactic object: a freely generated preadditive category whose objects are indexed by the integers and whose generating arrows are $d_n:n\to n-1$$d_n:n\to n-1$, subject only to the relation $d_{n-1}{d}_n=0$$d_{n-1}d_n=0$. This led to the formulation of chain complexes as additive functors from this syntactic category into a target preadditive category.

A second major contribution was Marco's insistence that cycles, boundaries, and homology should not be introduced merely by degreewise definitions followed by separate functoriality checks. Instead, Marco proposed that these should be obtained directly inside a functor category. This shifted the project from a standard explanation of chain complexes toward a genuinely functorial reconstruction of homology.

Marco further introduced the idea of viewing the differential as a universal natural transformation in the syntactic category. This observation led to the formulation of a natural transformation from the identity functor to the syntactic shift functor. Under any model, this universal syntactic differential becomes the ordinary differential of the corresponding chain complex.

Marco also raised the essential size issue: when considering the additive endofunctor category of $\mathsf{Ch}(R)$$\mathsf{Ch}(R)$ or $\mathsf{Ch}(\mathcal{A})$$\mathsf{Ch}(\mathcal A)$, one must be careful about whether the source category is small. This led to the inclusion of a Grothendieck universe convention in the final version, making the use of endofunctor categories mathematically controlled.

In short, Marco's main contributions were conceptual and structural:

1. identifying the preadditive sketch as the minimal syntax of chain complexes;

2. distinguishing preadditive structure from additive structure;

3. recognizing the universal differential inside the syntactic category;

4. demanding a fully functorial construction of $Z$$Z$, $B$$B$, and $H$$H$ inside an endofunctor category;

5. identifying the need for a universe convention when forming large functor categories;

6. integrating these ideas into a coherent mathematical narrative.

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The Assistant's Contributions

The assistant's role was primarily formal, corrective, and expository.

First, the assistant helped formulate the syntactic category explicitly. This included clarifying that the category is preadditive rather than additive, that its Hom groups are freely generated subject to the relation $d_{n-1}{d}_n=0$$d_{n-1}d_n=0$, and that models are additive functors into a target preadditive category.

Second, the assistant helped express the equivalence between chain complexes and additive functors:

$\mathsf{Ch}(\mathcal{A})\simeq \mathsf{Add}({\mathcal{C}}_{\mathrm{ch}},\mathcal{A})$$\mathsf{Ch}(\mathcal A)\simeq \mathsf{Add}(\mathcal C_{\mathrm{ch}},\mathcal A)$.

This included identifying chain maps with natural transformations between additive functors.

Third, the assistant clarified why $\mathsf{Ch}(\mathcal{A})$$\mathsf{Ch}(\mathcal A)$ is an abelian category when $\mathcal{A}$$\mathcal A$ is abelian. The argument was reformulated as a general result about additive functor categories: if the source is a small preadditive category and the target is abelian, then the additive functor category is abelian, with kernels, cokernels, images, and coimages computed pointwise.

Fourth, the assistant helped analyze the shift functor and the universal differential. In particular, an important correction was made regarding the signed shift. Initially, there was a risk of thinking that the family ${\eta }_n=d_n$$\eta_n=d_n$ would fail to define a natural transformation for the signed shift. The assistant checked the naturality condition and observed that both sides are zero because $d^2=0$$d^2=0$. Thus the universal differential works for the usual signed shift as well.

Fifth, the assistant helped formulate the construction of cycles, boundaries, and homology in the additive endofunctor category. The resulting definitions were:

$Z=\ker (\delta )$$Z=\ker(\delta)$,

$B=\mathrm{im}({\delta }^{-})$$B=\operatorname{im}(\delta^-)$,

$H=\mathrm{coker}(B\hookrightarrow Z)$$H=\operatorname{coker}(B\hookrightarrow Z)$.

This made the functoriality of $Z$$Z$, $B$$B$, and $H$$H$ automatic rather than something proved degree by degree.

Sixth, the assistant helped clarify the role of Grothendieck universes. The assistant explained that the additive endofunctor category of $\mathsf{Ch}(R)$$\mathsf{Ch}(R)$ is abelian only after an appropriate size convention is fixed, or else it should be read as a large abelian category with pointwise exact structure. This led to a cleaner and more defensible universe convention in the final text.

Finally, the assistant organized the material into a coherent English exposition, producing a Typora-ready draft with sections on the syntactic category, models, chain maps, abelian structure, shift functor, universal differential, induced model shift, additive endofunctor categories, cycles, boundaries, and homology.

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Nature of the Collaboration

The collaboration followed an iterative mathematical pattern.

Marco repeatedly supplied high-level structural insights, often in the form of compressed categorical intuitions. The assistant then expanded these intuitions into explicit definitions, checked technical details, identified hidden assumptions, and reorganized the argument into a stable written form.

Several important refinements emerged through this process:

1. The sketch should be preadditive, not necessarily additive.

2. The zero required for $d^2=0$$d^2=0$ is a zero morphism, not a zero object.

3. The category of chain complexes is best seen as an additive functor category.

4. The differential is the interpretation of a universal syntactic natural transformation.

5. The signed shift still admits the universal differential because $d^2=0$$d^2=0$.

6. Cycles, boundaries, and homology can be defined in an additive endofunctor category.

7. The endofunctor category requires an explicit universe convention.

The collaboration was therefore not merely editorial. It involved the transformation of a conceptual observation into a precise categorical framework.

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Division of Intellectual Labor

Marco's role was that of the primary conceptual originator. He identified the central viewpoint and repeatedly pushed the project away from degreewise definitions and toward a syntactic-functorial reconstruction.

The assistant's role was that of a formalizing collaborator and technical editor. It tested the definitions, corrected sign and size issues, supplied standard categorical facts where needed, and converted the emerging structure into a coherent written document.

The resulting note reflects both roles:

• the conceptual architecture comes from Marco's mathematical intuition;

• the formal stabilization and exposition were developed through interaction with the assistant.

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Outcome

The final result is a short categorical note showing that the ordinary theory of chain complexes can be reconstructed from a small preadditive syntactic category.

The main conclusion is that the chain complex axiom $d^2=0$$d^2=0$ is already encoded syntactically. Once this syntax is interpreted in an abelian category, the usual category of chain complexes, its abelian structure, the shift functor, the differential, cycles, boundaries, and homology all arise from functorial universal constructions.

The project therefore gives a compact example of how syntactic categories and model categories can clarify a familiar construction in homological algebra.

It shows that chain complexes are not merely sequences of objects and morphisms satisfying an equation. They are models of a preadditive syntax, and their homological constructions are induced by universal operations in functor categories.