Non-unital algebra objects of stable symmetric monoidal model categories by Smith ideal theory

TL;DR

This paper explores the relationship between non-unital and augmented unital algebra objects within stable symmetric monoidal model categories, utilizing Smith ideal theory to generalize their correspondence.

Contribution

It introduces a framework connecting non-unital algebra objects to augmented unital algebras via Smith ideal theory in stable symmetric monoidal model categories.

Findings

Established a generalized correspondence between non-unital and augmented algebra objects.

Applied Smith ideal theory to stable symmetric monoidal model categories.

Provided a new perspective on algebra object structures in homotopical algebra.

Abstract

This note remarks that the correspondence between non-unital algebras and augmented unital algebras can be derived from Hovey's Smith ideal theory. Applying Smith ideal theory of stable symmetric monoidal model category, we formulate non-unital algebra objects of stable symmetric monoidal model categories and generalize the correspondence between non-unital algebra objects and augmented algebra objects.

Equations

$$(m,a)\cdot (n,b)=(mn,na+mb+ab)$$

$$V\oplus (-):{\mathrm{CAlg}}_V^{\mathrm{nu}}\rightleftarrows {\mathrm{CAlg}}_{V//V}:\mathrm{Ker}({ϵ}_{(-)}:-\to V)$$

$$f\otimes g:X_0\otimes Y_0\to X_1\otimes Y_1,$$

$$f\square g:(X_0\otimes Y_1){⨿}_{X_0\otimes Y_0}(X_1\otimes Y_0)\to X_1\otimes Y_1.$$

$$\mathrm{cok}:{\mathrm{Ar}}^{\square }(\mathcal{C})\ni (f:X\to Y)\mapsto (\mathrm{cok}(f):Y\to \mathrm{Coker}(f))\in {\mathrm{Ar}}^{\otimes }(\mathcal{C})$$

$$\ker (f)\square \mathrm{Ker}(g)\to \ker (f\otimes g)$$

$$\mathrm{cok}:{\mathrm{Ar}}^{\square }(\mathcal{C})\rightleftarrows {\mathrm{Ar}}^{\otimes }(\mathcal{C}):\ker$$

$$\mathrm{cok}:{\mathrm{Ar}}^{\mathrm{im}}(\mathcal{C})\rightleftarrows {\mathrm{Ar}}^{\mathrm{coim}}(\mathcal{C}):\ker .$$

$$\mathrm{cok}:{\mathrm{Ar}}^{\square }(\mathcal{C})\rightleftarrows {\mathrm{Ar}}^{\otimes }(\mathcal{C}):\ker$$

$$\mathrm{cok}:{\mathrm{CAlg}}^{\mathrm{nu}}(\mathcal{C})\to \mathrm{CAlg}(\mathcal{C}{)}_{V//V}:\ker .$$

$$X\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\mathrm{cok}:\mathrm{Ar}(\mathcal{M})\rightleftarrows \mathrm{Ar}(\mathcal{M}):\ker$$

Full text

1. Introduction

A ring without the assumption existence of a multiplicative unit $1$ is called a non-unital ring. Any unital ring is regarded as a non-unital ring by forgetting the existence of $1$. This paper fixes a base unital ring $V$ and let $\mathrm{CAlg}^{\rm nu}_{V}$ denote the category of non-unital $V$-algebras. An augmentation $\varepsilon_{A}:A\to V$ is a ring homomorphism that the unit homomorphism $V\to A$ is a section of $\varepsilon_{A}$, and a $V$-algebra with an augmentation is called an augmented $V$-algebra, denoting $\mathrm{CAlg}_{V//V}$ the category of augmented $V$-algebras. It is well-known that there exists a categorical equivalence between $\mathrm{CAlg}^{\rm nu}_{V}$ and $\mathrm{CAlg}_{V//V}$: For any commutative $V$-algebra $A$, the direct sum $V\oplus A$ has a canonical unital ring structure defined by

[TABLE]

for $m,\,n\in V$ and $a,\,b\in A$. It is easily checked that the functor the adjunction

[TABLE]

is a pair of categorical equivalences, where $\mathrm{Ker}(\varepsilon_{(-)}:-\to V)$ is the kernel functor of augmentations.

The main result of this paper is a model categorical analogue of the above categorical equivalence (Theorem 3.4): A stable model category is a model category whose homotopy category is triangulated. We generalize the definition of non-unital commutative algebras for stable symmetric monoidal model categories by using Hovey’s Smith ideal theory [Hov14] and the following theorem:

Theorem 1.1** ([Hov14] Theorem 4.3).**

Let $\mathcal{M}$ be a cofibrantly generated stable symmetric monoidal model category.

• •

The cokernel functor $\mathrm{cok}:\mathrm{Ar}(\mathcal{M})\to\mathrm{Ar}(\mathcal{M})$ is a left Quillen equivalence form the projective model structure to the injective model structure.

• •

If $\mathcal{M}$ is a cofibrantly generated symmetric monoidal stable model category, the cokernel functor is a Quillen equivalence from the model category of Smith ideals to the model category of monoid morphisms.

$\Box$

An advantage of Smith ideal theory of stable symmetric monoidal model categories is that it enables us to clarify that a symmetric monoidal model structure of non-unital algebra objects is the push-out products induced by ones.

2. Smith ideal theory of symmetric monoidal model categories

We explain Hovey [Hov14]’s Smith ideal theory of symmetric monoidal categories. This section assumes that categories are always pointed, and [math] denotes the zero object.

2.1. Smith ideal theory of pointed categories

For any symmetric monoidal category $\mathcal{C}$, let $\mathrm{CAlg}(\mathcal{C})$ denote the subcategory of $\mathcal{C}$ spanned by commutative monoidal objects in the sense of MacLane [Mac88].

Let $[1]$ denote the category with two objects [math] and $1$, and only one non-identity morphism $0\to 1$. For any pointed category $\mathcal{C}$, the diagram category $\mathrm{Fun}([1],\,\mathcal{C})$ is called the arrow category, and $\mathrm{Ar}(\mathcal{C})$ denotes it. The symmetric monoidal structure of $\mathcal{C}$ inherits $\mathrm{Ar}(\mathcal{C})$ two symmetric monoidal structures: For any two morphisms in $\mathcal{C}$, $f:X_{0}\to X_{1}$ and $g:Y_{0}\to Y_{1}$, the tensor product monoidal structure is defined by $f\otimes g$ as the composition

[TABLE]

and another is the push-out product monoidal structure defined by the induced morphism:

[TABLE]

We let $\mathrm{Ar}^{\otimes}(\mathcal{C})$ the arrow category with the tensor product monoidal structure and $\mathrm{Ar}^{\Box}(\mathcal{C})$ with the push-out monoidal structure. Hovey proved the cokernel functor is strongly monoidal and admits a lax monoidal right adjoint:

Theorem 2.1** ([Hov14] Theorem 1.4).**

Let $\mathcal{C}$ be a pointed closed symmetric monoidal category. Then the functor

[TABLE]

is strongly symmetric monoidal, and its right adjoint is the kernel functor $\mathrm{ker}:(f:X\to Y)\mapsto(\mathrm{ker}(f):\mathrm{Ker}(f)\to X)$. $\Box$

Hovey mentioned in [Hov14], in general, the kernel functor $\mathrm{ker}:\mathrm{Ar}(\mathcal{C})\to\mathrm{Ar}(\mathcal{C})$ is lax symmetric monoidal: A canonical functor

[TABLE]

is induced by the canonical morphism $\mathrm{cok}(\mathrm{ker}(f)\Box\mathrm{Ker}(g))\simeq\mathrm{cok}(\mathrm{ker}(f))\otimes\mathrm{cok}(\mathrm{ker}(g))\to f\otimes g$ for any $f,\,g\in\mathrm{Ar}^{\otimes}(\mathcal{C})$. Hence those functors $\mathrm{cok}$ and $\mathrm{ker}$ send monoid objects to monoid objects. A Smith ideal is a commutative monoid object of the symmetric monoidal category $\mathrm{Ar}(\mathcal{C})$ with respect to the push-out product monoidal structure.

2.2. Definition of commutative non-unital algebra objects of symmetric monoidal categories

For any pointed closed symmetric monoidal category $\mathcal{C}$ with a monoidal unit $V$, let $\mathrm{CAlg}(\mathcal{C})_{V//V}$ denote the full subcategory of $\mathrm{Ar}^{\otimes}(\mathcal{C})$ spanned by augmentations of commutative algebra objects of the tensor product monoidal structure.

Let $\mathcal{C}$ be a pointed closed symmetric monoidal category and $\mathrm{Ar}^{\rm im}(\mathcal{C})$ denotes the full subcategory spanned those objects $f:R\to S$ such that the unit $f\to\mathrm{ker}(\mathrm{cok}(f))$ is an isomorphism. Then the cokernel functor $\mathrm{Ar}^{\rm im}(\mathcal{C})\to\mathrm{Ar}^{\rm coim}(\mathcal{C})$ is a categorical equivalence.

Proposition 2.2**.**

Let $\mathcal{C}$ be a locally presentable abelian symmetric monoidal category. Then the arrow category $\mathrm{Ar}(\mathcal{C})$ is also locally presentable and there exist reflective localization functors $L_{\rm im}:\mathrm{Ar}^{\Box}(\mathcal{C})\to\mathrm{Ar}^{\rm im}(\mathcal{C})$ by the unit $\mathrm{id}\to\mathrm{im}$ and $L_{\rm coim}:\mathrm{Ar}^{\otimes}(\mathcal{M})\to\mathrm{Ar}^{\rm coim}(\mathcal{M})$ by the counit $\mathrm{coim}\to\mathrm{id}$ such that the adjunction

[TABLE]

induces categorical equivalences

[TABLE]

• proof.

Since any abelian category is binormal, $f\to\mathrm{im}(f)$ is an isomorphism if and only if $f$ is a monomorphism. By Adámek–Rosický [AR94, p.44, Corollary 1.5.4], the arrow category is locally presentable, and it admits reflective localization. By the definitions of those functors: $\mathrm{im}=\mathrm{ker}\circ\mathrm{cok}$ and $\mathrm{coim}=\mathrm{cok}\circ\mathrm{ker}$, the restriction $\mathrm{cok}:\mathrm{Ar}^{\rm im}(\mathcal{C})\to\mathrm{Ar}^{\rm coim}(\mathcal{C})$ are quasi-inverse functors. $\Box$

Definition 2.3**.**

Let $\mathcal{C}$ be a locally presentable symmetric monoidal abelian category with a monoidal unit $V$. A commutative non-unital monoid object of $\mathcal{C}$ is a Smith ideal $j:I\to R$ in the localized full subcategory $\mathrm{Ar}^{\rm im}(\mathcal{C})$ satisfying that the cokernel $\mathrm{cok}(j):R\to\mathrm{Coker}(j)$ is isomorphic with an argumentation $\varepsilon_{R}:R\to V$, and $\mathrm{CAlg}^{\rm nu}(\mathcal{C})$ denote the full subcategory of $\mathrm{CAlg}(\mathrm{Ar}^{\Box}(\mathcal{C}))$ spanned by non-unital commutative algebra objects, where $\mathrm{Ar}^{\Box}(\mathcal{C})$ denotes the arrow category of $\mathcal{C}$ whose monoidal structure is the push-out monoidal structure.

Proposition 2.4**.**

Further, let $\mathrm{CAlg}^{\rm nu}(\mathcal{C})$ denote the full subcategory of $\mathrm{Ar}^{\Box}(\mathcal{C})$ the full subcategory which is the essential image of the restriction of the kernel functor on $\mathrm{CAlg}(\mathcal{C})_{V//V}$. Then any object $j:I\to R$ of $\mathrm{CAlg}^{\rm nu}(\mathcal{C})$, the unit $j\to\mathrm{im}(j)$ is an isomorphism.

• proof.

By definition of the category $\mathrm{CAlg}^{\rm nu}(\mathcal{C})$, one has an isomorphism $V\oplus I\to R$. The assertion is clear. $\Box$

Corollary 2.5**.**

Let $\mathcal{C}$ be a locally presentable symmetric monoidal abelian model category. The adjunction

[TABLE]

induces categorical equivalences

[TABLE]

$\Box$

3. Non-unital commutative algebra objects of symmetric monoidal model categories

A symmetric monoidal model category $\mathcal{M}$ is a model category with a symmetric monoidal structure $-\otimes-:\mathcal{M}\times\mathcal{M}\to\mathcal{M}$ such that, for any object $M$ of $\mathcal{M}$, those functors $(-)\otimes M$ and $M\otimes(-)$ are left Quillen functors on $\mathcal{M}$.

3.1. The arrow categories of pointed symmetric monoidal model categories

The category $\mathrm{Ar}(\mathcal{M})$ has two canonical model structures, the injective model structure and the projective model structure induced by $\mathcal{M}$’s:

Definition 3.1**.**

Let $\mathcal{M}$ be a model category. The arrow category $\mathrm{Ar}(\mathcal{M})$ has the following two model structures.

• •

(Injective model structure) A morphism $\alpha:(f:X_{0}\to X_{1})\to(g:Y_{0}\to Y_{1})$ is a cofibrations (resp. weak equivalence) in $\mathrm{Ar}(\mathcal{M})$ if and only if so is each $\mathrm{Ev}_{i}(\alpha)$ for $i=0,\,1$. Fibrations are morphisms with the right lifting property for all trivial cofibrations.

• •

(Projective model structure) A morphism $\alpha:(f:X_{0}\to X_{1})\to(g:Y_{0}\to Y_{1})$ is a fibrations (resp. weak equivalence) in $\mathrm{Ar}(\mathcal{M})$ if and only if so is each $\mathrm{Ev}_{i}(\alpha)$ for $i=0,\,1$. Cofibrations are morphisms with the right lifting property for all trivial fibrations.

In a pointed model category $\mathcal{M}$, we consider a homotopically commutative diagram:

[TABLE]

If the diagram is a homotopy Cartesian square, then $X$ is said to be a homotopy kernel of $g$, and if it is a homotopy coCartesian square, then $Z$ is a homotopy cokernel of $f$. We can consider homotopy image objects and homotopy coimage objects as additive categories.

On the arrow category $\mathrm{Ar}(\mathcal{M})$, those functors $\mathrm{cok}:\mathrm{Ar}(\mathcal{M})\to\mathrm{Ar}(\mathcal{M})$ and $\mathrm{ker}:\mathrm{Ar}^{\otimes}(\mathcal{M})\to\mathrm{Ar}^{\Box}(\mathcal{M})$ are defined as follows: For a morphism $f:X\to Y$ in $\mathcal{M}$, the arrow $\mathrm{cok}(f)$ is $Y\to\mathrm{Coker}f$ and $\mathrm{ker}(f)$ $\mathrm{Ker}(f)\to X$. Then the pair

[TABLE]

is a Quillen adjunction.

Definition 3.2**.**

Let $\mathcal{M}$ be a pointed symmetric monoidal model category. A Smith ideal in $\mathcal{M}$ is a monoid object $j:I\to R$ in the symmetric monoidal model category $\mathrm{Ar}^{\Box}(\mathcal{M})$ with respect to the push-out product monoidal model structure.

We say that a Smith ideal $j:I\to R$ is unit cokernel if the cokernel of $j$ is isomorphic to the monoidal unit object $V$ and $\mathrm{cok}(j):R\to\mathrm{Coker}(j)$ is an augmentation of the unit morphism $V\to R$. Let $\mathrm{CAlg}^{\rm nu}(\mathcal{C})$ denote the full subcategory of $\mathrm{CAlg}(\mathrm{Ar}^{\Box}(\mathcal{C}))$ spanned by unit cokernel Smith ideals.

Definition 3.3**.**

Let $\mathcal{M}$ be a stable symmetric monoidal model category with a monoidal unit $V$ and $\mathrm{CAlg}^{\rm nu}(\mathcal{M})$ denote the full subcategory $\mathrm{CAlg}(\mathrm{Ar}^{\Box}(\mathcal{M}))$ spanned by Smith ideals whose cokernels are weakly equivalent to $V$. We say that an object of $\mathrm{CAlg}^{\rm nu}(\mathcal{M})$ is a non-unital commutative algebra object of $\mathcal{M}$.

Theorem 3.4**.**

Let $\mathcal{M}$ be a stable symmetric monoidal model category with a monoidal unit $V$. Then the Quillen equivalence

[TABLE]

induces a left Quillen equivalence between $\mathrm{cok}:\mathrm{CAlg}^{\rm nu}(\mathcal{M})\to\mathrm{CAlg}(\mathcal{M})_{V//V}$.

• proof.

By using the second statement of Theorem 1.1, the cokernel functor $\mathrm{cok}:\mathrm{Ar}(\mathcal{M})\to\mathrm{Ar}(\mathcal{M})$ induces the left Quillen functor $\mathrm{cok}:\mathrm{CAlg}^{\rm nu}(\mathcal{M})\to\mathrm{CAlg}(\mathcal{M})_{V//V}$, being essentially surjective on the homotopy categories by definition. Since the unit $u:\mathrm{Id}_{\mathrm{Ar}(\mathcal{M})}\to\mathrm{ker}\circ\mathrm{cok}$ is a weak equivalence by Theorem 1.1 again, the induced functor $\mathrm{cok}:\mathrm{CAlg}^{\rm nu}(\mathcal{M})\to\mathrm{CAlg}(\mathcal{M})_{V//V}$ is homotopically fully faithful. $\Box$

Remark 3.5*.*

If $\mathcal{M}$ is a locally presentable stable symmetric monoidal model category, the presentable $\infty$-category of augmented algebra is represented by the model category $\mathrm{CAlg}(\mathcal{M})_{V//V}$. Hence the left-hand-side $\mathrm{CAlg}^{\rm nu}(\mathcal{M})$ is equivalent to the $\infty$-category defined by Lurie [Lur17, p.949, Definition 5.4.4.9 and Proposition 5.4.4.10].