How do these definitions of factorization algebra compare?

Question

Question

Several sources define (homotopy) factorization algebras in a seemingly different manner (I am looking at [CG], [Gi], and [CFM].) I wish to know how they compare with each other.

I apologize for the lengthy post. If you are familiar with factorization algebra, you can probably skip the section entitled "Preliminary Definitions."

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Preliminary Definitions

To be absolutely clear, let us start from the definition of prefactorization algebras. For definiteness, I will consider the case where the target category is the category $\mathcal{C}=\mathsf{Ch}(\mathcal{A})$ of cochain complexes over a Grothendieck abelian category $\mathcal{A}$. Let $M$ be an $n$-manifold.

Definition(Prefactorization algebra) A prefactorization algebra on $M$ (with values in $\mathcal{C}$) consists of the following data:

• For each open subset $U\subset M$, a cochain complex $FU\in\mathcal{C}$,

• For each collection $U_{1},\dots,U_{k},V\subset M$ of open subsets of $M$ such that $U_{i}\cap U_{j}=\emptyset$ and $\bigcup_{i}U_{i}\subset V$, a map $$ m_{U_{1},\dots,U_{k};V}:\bigotimes_{i=1}^{k}FU_{i}\to FV. $$

These data are required to satisfy the following conditions:

• For each open set $U\subset M$, the map $m_{U,U}$ is the identity map.

• For each collection of open sets $U_{11},\dots,U_{k_{1}1},\dots,U_{1l},\dots,U_{k_{l}l},V_{1},\dots,V_{l},W\subset M$ such that $U_{ij}\cap U_{i'j}=\emptyset$, $V_{j}\cap V_{j'}=\emptyset$, and $\bigcup_{i}U_{ij}\subset V_{j}$ and $\bigcup_{j}V_{j}\subset W$, we have $$ m_{U_{11},\dots,U_{k_{l}l};W}=m_{V_{1},\dots,V_{l};W}\circ\bigotimes_{j=1}^{l}m_{U_{1j},\dots,U_{k_{j}j};V_{j}}. $$

To state various definitions of factorization algebras, we also need some definitions on covers.

Definition(Weiss cover and factorizing cover) Let $\mathcal{U}$ be an open cover of $M$.

• We call $\mathcal{U}$ a Weiss cover if for each finite set $S\subset M$, there is an element $U\in\mathcal{U}$ such that $S\subset U$.

• We call $\mathcal{U}$ a factorizing cover if for each finite subset $S\subset M$, there is a finite subset $\alpha\subset\mathcal{U}$ of pairwise disjoint open sets such that $S\subset\bigcup_{V\in\alpha}V$.

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Definitions of Factorization Algebra

We now look at the definitions of (homotopy) factorization algebras in the sources cited above. Let $F$ be a prefactorization algebra on $M$.

Definition (1), found in [CG, 6.1.4]. We call $F$ a factorization algebra if for each Weiss cover $\mathcal{U}$ of an open set $U\subset M$, the map $$ T\to FU $$ is a quasi-isomorphism, where $T$ denotes the totalization of the normalized chain complex in $\cal C$ (regarded as a cochain complex in $\cal C$ concentrated in negative degrees) associated with the simplicial object $\{\bigoplus_{U_{0},\dots,U_{n}\in\mathcal{U}}F\left(U_{0}\cap\cdots\cap U_{n}\right)\}_{n\geq0}$.

Definition (2), found in [Gi, p. 36]. We call $F$ a factorization algebra if for each factorizing cover $\cal U$ of an open set $U\subset M$, the map $$ T\to FU $$ is a quasi-isomorphism, where the object $T\in\cal C$ is defined in the following manner: Let $P\mathcal{U}$ denote the set of finite subsets $\alpha\subset\mathcal{U}$ consisting of pairwise disjoint open sets. Given elements $\alpha_{0},\dots,\alpha_{n}\in P\mathcal{U}$, we will write $$ F\left(\alpha_{0},\dots,\alpha_{n}\right)=\bigotimes_{U_{j}\in\alpha_{j}}F\left(\bigcap_{j=0}^{n}U_{j}\right). $$ The objects $\{\bigoplus_{\alpha_{0},\dots,\alpha_{n}\in P\mathcal{U}}F(\alpha_{0},\dots,\alpha_{n})\}_{n\geq0}$ form a simplicial objet in $\cal C$. The object $T\in\cal C$ is the totalization of the Moore complex of this simplicial object (regarded as a cochain complex in $\cal C$ concentrated in negative degrees).

Definition (3), found in [CFM, Definition 2.2]. We call $F$ a factorization algebra if for each open set $U\subset M$ and each Weiss cover $\mathcal{U}$ of $U$, the map $$ \operatorname{hocolim}_{\alpha\subset\mathcal{U}\text{ finite}}F\left(\bigcap_{V\in\alpha}V\right)\to FU $$ is a quasi-isomorphism, and moreover for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, the map $m_{U_{1},\dots,U_{k};\bigcup_{i}U_{i}}$ is a quasi-isomorphism.

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Comments

I am unfamiliar with the computation of homotopy colimits of cochain complexes, but looking at this MO question, I suspect that the object $T$ in Definitions (1) and (2) are homotopy colimits of the corresponding simplicial object. So they are similar in the sense that they both use simplicial objects, but they differ in their choices of covers.

I am perplexed by the sudden appearance of the simplicial objects in these definitions; perhaps knowing something about homotopy colimits of cochain complexes might help, but the only reference I have on that topic is [CG], which I found not so helpful in explaining why the simplicial objects in (1) and (2) appeared.)

Definition (3) looks like a hybrid of (1) and (2) because it uses Weiss covers and somehow tries to incorporate the information of multiplicative structures of $F$ in the definition.

All these definitions look different to me.

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References

[CG] Costello, K., Gwilliam, O. (2017). Factorization Algebras in Quantum Field Theory: Volume 1.

[CFM] Carmona,V., Flores,R., and Fernando Muro. (2022). A model structure for locally constant factorization algebras. arxiv.2107.14174

[Gi] Grégory Ginot. (2014). Notes on factorization algebras, factorization homology and applications. arxiv.1307.5213

Answer 1

I believe that Definition 2 and Definition 3 are equivalent. This involves that Definition 2 implies that F is multiplicative ("for each pairwise disjoint open sets $U_{1},\dots,U_{k}\subset M$, the map $m_{U_{1},\dots,U_{k};\bigcup_{i}U_{i}}$ is a quasi-isomorphism"), that every Weiss cover can be refined by a cover consisting of disjoint unions of small balls, and that (as you say) that there is a relationship between homotopy colimits and simplicial objects.

(In a cocomplete category every colimit can be written as a specific coequalizer of two coproducts. In an \infty-category like the derived \infty-category of a field, this generalizes to a colimit over a simplicial object. There is an appendix in [CG] which explains parts of this story and contains some references - not sure if you already saw this and wanted more or not.)

Definition 1 + multiplicative is equivalent to Definition 3.

The examples in [CG] satisfy Definition 1 but most of them are not multiplicative for $A$ the category of vector spaces with the ordinary tensor product of vector spaces. Therefore Definition 1 is not equivalent to Definition 3. These examples are multiplicative if one uses a completed versions of the tensor product, but then one would need to do homological algebra with some flavor of complete vector spaces which usually do not form an abelian category. Maybe what one should do is use the symmetric monoidal category of liquid vector spaces, because they have a good tensor product and form an abelian category at the same time.

Answer 2

OK, here is the full story, which confirms what is written in Daniel Bruegmann's answer. In what follows, I will work with a fixed prefactorization algebra $F$ and an $n$-manifold $M$. I will make use of the fact that the homotopy colimits of simplicial (co)chain complexes are given by the (direct sum) totalization of the associated double complex; see my note [Ara] for a detailed account on this. Our argument will also make use of the fact that reindexing by homotopy final functors preserves homotopy colimits; [ArØr] is a good reference for this.

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A FEW REMARKS

• Definition (2) does not quite make sense. Indeed, as soon as $\mathcal{U}$ contains some element $U$, we would need a map $I=F(\emptyset,\{U\})\to F(\{U\})$ corresponding to the face map $d_{0}$; but I don't know where such a map comes from (except when $F(\emptyset)=I$). I will therefore redefine $F(\alpha_{0},\dots,\alpha_{n})$ in Definition (2) as $$ F(\alpha_{0},\dots,\alpha_{n})=F\left(\bigcap_{i=0}^{n}O(\alpha_{i})\right), $$ where we set $O(\alpha)=\bigcup_{V\in\alpha}V$ for each $\alpha\in P\mathcal{U}$.

• In Definition (3), we should index the homotopy colimits by {}nonempty{} subsets of $\mathcal{U}$. Otherwise, the homotopy colimit will just be $FU$.

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THE EQUIVALENCES

We will prove the following:

Proposition 1. Let $F$ be a prefactorization algebra on $M$. Suppose that, for each finite set $U_{1},\dots,U_{k}$ of open sets of $M$, the map $\bigotimes_{i=1}^{k}F(U_{i})\to F(\bigcup_{i=1}^{k}U_{i})$ is a quasi-isomorphism. Then the following conditions are equivalent:

(A) $F$ is a factorization algebra in the sense of Definition (1).

(B) $F$ is a factorization algebra in the sense of Definition (2) (or rather, its modified version explained in the above remark)

(C) $F$ is a factorization algebra in the sense of Definition (3).

(D) Let $\mathcal{U}$ be a Weiss cover of an open set $U\subset M$ such that, for each $V\in\mathcal{U}$, every open subset of $V$ belongs to $\mathcal{U}$. Then the map $$ \operatorname{hocolim}_{V\in\mathcal{U}}F(V)\to F(U) $$ is a quasi-isomorphism.

We need a lemma:

Lemma 2. Let $S$ be a nonempty set. Let $\mathcal{X}(S)$ denote the category whose objects are the finite nonempty sequences $\left(x_{0},\dots,x_{n}\right)$ of elements of $S$, and whose morphism $\left(x_{0},\dots,x_{n}\right)\to\left(y_{0},\dots,y_{m}\right)$ is a poset map $u:[m]\to[n]$ such that $x_{u(i)}=y_{i}$ for every $0\leq i\leq m$. The nerve of $\mathcal{X}(S)$ is weakly contractible.

(Proof of Lemma 2.) The category $\mathcal{X}(S)$ is the category of elements of the functor $S^{[\bullet]}:\mathbf{\Delta}^{\mathrm{op}}\to\mathsf{Set}$ defined by $[n]\mapsto S^{[n]}$. By Thomason's homotopy colimit theorem, the simplicial set $N(\mathcal{C})$ has the weak homotopy type of the homotopy colimit of the composite $\mathbf{\Delta}^{\mathrm{op}}\xrightarrow{S^{[\bullet]}}\mathsf{Set}\hookrightarrow\mathsf{sSet}$. The latter is given by the diagonal of the corresponding bisimplicial set, which, in this case, is the simplicial set $S^{[\bullet]}$. It will therefore suffice to show that the simplicial set $S^{[\bullet]}$ is weakly contractible. We claim that $S^{[\bullet]}$ is a contractible Kan complex. Observe that, given a simplicial set $X$, defining a map of simplicial sets $X\to S^{[\bullet]}$ is equivalent to defining a set map $X_{0}\to S$. So the simplicial set has the extension property for the inclusion $\partial\Delta^{n}\subset\Delta^{n}$ for every $n>0$. Since $S$ is nonempty, the simplicial set $S^{[\bullet]}$ has the extension property for the inclusion $\partial\Delta^{0}\subset\Delta^{0}$. $\blacksquare$

(Proof of Proposition 1.)

(A)$\implies$(D): Let $\mathcal{U}$ be as in (D). The simplicial object $\left\{ \bigoplus_{U_{0},\dots,U_{n}\in\mathcal{U}}F\left(\bigcap_{i=0}^{n}U_{i}\right)\right\} _{n\geq0}$ is a left Kan extension of the composite $\mathcal{X}(\mathcal{U})\xrightarrow{\varphi}\mathcal{U}\xrightarrow{F}\mathcal{C}$ along the projection $\mathcal{X}\to\mathbf{\Delta}^{\mathrm{op}}$, where the functor $\varphi$ is given by $\varphi(U_{0},\dots,U_{n})=\bigcap_{i=0}^{n}U_{i}$. Therefore, condition (A) implies that the map $\operatorname{hocolim}_{(U_{0},\dots,U_{n})\in\mathcal{X}}F(\bigcap_{i=0}^{n}U_{i})\to FU$ is a quasi-isomorphism. It will therefore suffice to show that $\varphi$ is homotopy final, which follows from Lemma 2.

(D)$\implies$(A): Let $\mathcal{U}$ be a Weiss cover of an open set $U\subset M$. Arguing as in the previous paragraph, it will suffice to show that the map $\operatorname{hocolim}_{(U_{0},\dots,U_{n})\in\mathcal{X}}F(\bigcap_{i=0}^{n}U_{i})\to FU$ is a quasi-isomorphism. Let $\overline{\mathcal{U}}$ denote the set of open subsets of elements of $\mathcal{U}$, and define $\varphi:\mathcal{X}(\mathcal{U})\to\overline{\mathcal{U}}$ as in the previous paragraph. By Lemma 2, the functor $\varphi$ is homotopy final, so condition (D) implies that the map $\operatorname{hocolim}_{(U_{0},\dots,U_{n})\in\mathcal{X}}F(\bigcap_{i=0}^{n}U_{i})\to FU$ is a quasi-isomorphism, as claimed.

(D)$\implies$(C): Let $\mathcal{U}$ be a Weiss cover of an open set $U\subset M$. Let $P$ denote the set of finite nonempty subsets of $\mathcal{U}$, and let $\overline{\mathcal{U}}$ denote the set of open subsets of elements of $\mathcal{U}$. The assignment $\alpha\mapsto\bigcap_{V\in\alpha}V$ determines a map $\varphi:P^{\mathrm{op}}\to\overline{\mathcal{U}}$. To complete the proof, it suffices to show that $\varphi$ is homotopy final, which is trivial.

(C)$\implies$(D): This is similar to (C)$\implies$(D).

(B)$\implies$(D): Let $\mathcal{U}$ be as in (D). Arguing as in the proof of (A)$\implies$(D) (and noting that $\mathcal{U}$ is a factorizing cover), we are reduced to showing that the functor $\mathcal{X}(P\mathcal{U})\to\mathcal{U}$ defined by $(\alpha_{0},\dots,\alpha_{n})\mapsto\bigcap_{i=0}^{n}O(\alpha_{i})$ is homotopy final. This follows from Lemma 2.

(D)$\implies$(B): Let $\mathcal{U}$ be a factorizing cover of an open set $U\subset M$. Let $\mathcal{V}$ denote the set of all open subsets of the topological spaces $\{U(\alpha)\}_{\alpha\in P\mathcal{U}}$. Note that $\mathcal{V}$ is a Weiss cover. Arguing as in the proof of (D)$\implies$(A), we are reduced to showing that the functor $\mathcal{X}(P\mathcal{U})\to\mathcal{V}$ defined by $(\alpha_{0},\dots,\alpha_{n})\mapsto\bigcap_{i=0}^{n}O(\alpha_{i})$ is homotopy final. This is a consequence of Lemma 2. $\blacksquare$

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REFERENCES

[Ara] Arakawa, K. (2023). Homotopy Limits and Homotopy Colimits of Chain Complexes. arxiv.2310.00201

[ArØr] Arkhipov, S., Ørsted, S. Homotopy (co)limits via homotopy (co)ends in general combinatorial model categories. arxiv.1807.03266