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

Yuki Kato National institute of technology, Ube college, 2-14-1, Tokiwadai, Ube, Yamaguchi, JAPAN 755-8555. [email protected]

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.

Key words and phrases:

Symmetric monoidal model categories, Smith ideals, Non-unital algebras

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 $\mathrm{Alg}^{\rm nu}_{V}$ 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 $\mathrm{Alg}_{V//V}$ the category of augmented $V$ -algebras. It is well-known that there exists a categorical equivalence between $\mathrm{Alg}^{\rm nu}_{V}$ and $\mathrm{Alg}_{V//V}$ : For any $V$ -algebra $A$ , the direct sum $V\oplus A$ has a canonical unital ring structure defined by

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

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

V\oplus(-):\mathrm{Alg}^{\rm nu}_{V}\rightleftarrows\mathrm{Alg}_{V//V}:\mathrm{Ker}(\epsilon_{(-)}:-\to V)

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 algebras for stable symmetric monoidal model categories by using Hovey’s Smith ideal theory [Hov14, Theorem 4.3]. 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.

Acknowledgements

The author was supported by Grants-in-Aid for Scientific Research No.23K03080, Japan Society for the Promotion of Science.

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 $0$ denotes the zero object.

2.1. Smith ideal theory of pointed categories

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

Let $[1]$ denote the category with two objects $0$ 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

f\otimes g:X_{0}\otimes Y_{0}\to X_{1}\otimes Y_{1},

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

f\Box g:(X_{0}\otimes Y_{1})\amalg_{X_{0}\otimes Y_{0}}(X_{1}\otimes Y_{0})\to X_{1}\otimes Y_{1}.

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

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

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

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

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 monoid object of the symmetric monoidal category $\mathrm{Ar}(\mathcal{C})$ with respect to the push-out product monoidal structure.

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

For any pointed closed symmetric monoidal category $\mathcal{C}$ with a monoidal unit $V$ , let $\mathrm{Alg}(\mathcal{C})_{V//V}$ denote the full subcategory of $\mathrm{Ar}^{\otimes}(\mathcal{C})$ spanned by augmentations of 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

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

induces categorical equivalences

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

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 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 $\mathrm{Alg}^{\rm nu}(\mathcal{C})$ denote the full subcategory of $\mathrm{Alg}(\mathrm{Ar}^{\Box}(\mathcal{C}))$ spanned by non-unital 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 $\mathrm{Alg}^{\rm nu}(\mathcal{C})$ 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 $\mathrm{Alg}(\mathcal{C})_{V//V}$ . Then any object $j:I\to R$ of $\mathrm{Alg}^{\rm nu}(\mathcal{C})$ , the unit $j\to\mathrm{im}(j)$ is an isomorphism.

proof.

By definition of the category $\mathrm{Alg}^{\rm nu}(\mathcal{C})$ , 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

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

induces categorical equivalences

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

$\Box$

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:

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

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

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 $\mathrm{Alg}^{\rm nu}(\mathcal{C})$ denote the full subcategory of $\mathrm{Alg}(\mathrm{Ar}^{\Box}(\mathcal{C}))$ 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 $\mathrm{Alg}^{\rm nu}(\mathcal{M})$ denote the full subcategory $\mathrm{Alg}(\mathrm{Ar}^{\Box}(\mathcal{M}))$ spanned by Smith ideals whose cokernels are weakly equivalent to $V$ . We say that an object of $\mathrm{Alg}^{\rm nu}(\mathcal{M})$ 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

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

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

proof.

By using the second statement of [Hov14, Theorem 4.3], the cokernel functor $\mathrm{cok}:\mathrm{Ar}(\mathcal{M})\to\mathrm{Ar}(\mathcal{M})$ induces the left Quillen functor $\mathrm{cok}:\mathrm{Alg}^{\rm nu}(\mathcal{M})\to\mathrm{Alg}(\mathcal{M})_{V//V}$ , 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 [Hov14, Theorem 4.3] again, the induced functor $\mathrm{cok}:\mathrm{Alg}^{\rm nu}(\mathcal{M})\to\mathrm{Alg}(\mathcal{M})_{V//V}$ 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 $\mathrm{Alg}(\mathcal{M})_{V//V}$ . Hence the left-hand-side $\mathrm{Alg}^{\rm nu}(\mathcal{M})$ is equivalent to the $\infty$ -category defined by Lurie [Lur17, p.949, Definition 5.4.4.9 and Proposition 5.4.4.10].

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