##
**Flows revisited: the model category structure and its left determinedness.**
*(English.
French summary)*
Zbl 1452.18010

A flow \(X\) is roughly speaking a small category without identities in which the set of morphisms \(Mor X\) is endowed with the structure of a topological space, and the set of objects \(Ob X\) is considered as a discrete space, and the composition
\[
\star: Mor X\times_{Ob X}Mor X \to Mor X
\] is continuous.
The flows were introduced for mathematical modeling of parallel processes
by P. Gaucher [Homology Homotopy Appl. 5, No. 1, 549–599 (2003; Zbl 1069.55008)].
The following notations are used: \({\mathbb P} X:= Mor X\), \(X^0:=Ob X\).
Morphisms between flows \(f: X\to X'\) are functors such that mappings
\({\mathbb P} f: {\mathbb P} X\to {\mathbb P} X'\) are continuous.
The category of flows and their morphisms is denoted by \({\text{ Flow}}\).

The category \({\text{ Top}}\) denotes a bicomplete locally presentable cartesian closed full subcategory of the category of general topological spaces containing all CW-complexes.

For example, for any topological space \(X\), we denote by \(Glob(X)\) the flow with \({\mathbb P} Glob(X)=X\), \(Glob(X)^0= \{0, 1\}\) and morphisms \(0 \xrightarrow{x} 1\) defined for all \(x\in X\). This induces a functor from \({\text{ Top}}\) to the category \({\text{ Flow}}\).

The paper under review continues the research of its author, published in papers [Theory Appl. Categ. 16, 59–83 (2006; Zbl 1088.55015); Int. J. Math. Math. Sci. 2007, Article ID 87404, 20 p. (2007; Zbl 1149.55009); New York J. Math. 12, 319–348 (2006; Zbl 1109.55010); New York J. Math. 12, 63–95 (2006; Zbl 1110.55011); Appl. Categ. Struct. 13, No. 5–6, 371–388 (2005; Zbl 1115.68103); Homology Homotopy Appl. 7, No. 1, 51–76 (2005; Zbl 1085.55003)].

Section 2 describes Isaev’s approach to constructing model categories. A locally representable category \(\mathcal K\) is considered. According to [J. Rosický, Appl. Categ. Struct. 17, No. 3, 303–316 (2009; Zbl 1175.55013)], combinatorial model categories were introduced as model categories which are locally presentable and cofibrantly generated. The paper under review cites source [A. Joyal, Notes on quasicategories. available at http://preview.tinyurl.com/o6bkdv8, Proposition E.1.10] leading to the statement that for a given set of maps \(I\), there exists at most one combinatorial model category structure on \({\mathcal K}\) such that the set of generating cofibrations is \(I\) and such that all objects are fibrant. The required results from Isaev’s paper [Theory Appl. Categ. 33, 43–66 (2018; Zbl 1393.55012)] on fibrant objects applied to a locally presentable category are collected in Proposition 2.1.

For \(n\geqslant 1\), denote by \(D^n\) the \(n\)-dimensional disk, and by \(S^{n-1}\) the \((n-1)\)-dimensional sphere. Let \(D^0=\{0\}\) and \(S^{-1}= \emptyset\). Denote by \(C\) the inclusion \(\emptyset \to \{0\}\) and by \(R\) the unique map \(\{0, 1\}\to \{0\}\). Let \[ I^{gl}_+= \{Glob(S^{n-1})\subset Glob(D^n)\}| n\geq 0\} \cup\{ C: \emptyset \to \{0\}, R:\{0, 1\}\to \{0\}\} \] be the set of morphisms in the category \({\text{ Flow}}\).

Theorem 3.11. There exists a unique model category structure on \({\text{ Flow}}\) such that \(I^{gl}_+\) is the set of generating cofibrations and such that all objects are fibrant.

Section 4 is devoted to left determinedness of the model category of flows.

For \({\mathcal C}\) a category, a class \(K\subset Mor({\mathcal C})\) is said to satisfy 2-out-of-3 property if for all composable \(f, g\in Mor({\mathcal C})\) we have that if two of the three morphisms \(f, g\) and the composite \(g\circ f\) is in \(K\), then so is the third.

Definition 4.1. Let \(I\) be a set of maps of a locally presentable category \({\mathcal K}\). A class of maps \(\mathcal W\) is a localizer (with respect to \(I\)) or an \(I\)-localizer if \({\mathcal W}\) satisfies:

– Every map satisfying the RLP with respect to the maps of \(I\) belongs to \({\mathcal W}\).

– \({\mathcal W}\) is closed under retract and satisfies the 2-out-of-3 property.

– The class of maps \(cof(I)\cap {\mathcal W}\) is closed under pushout and transfinite composition.

In particular, the class of all maps is an \(I\)-localizer. The class of \(I\)-localizers is closed under arbitrarily large intersection. Therefore there exists a smallest \(I\)-localizer denoted by \({\mathcal W}_I\).

In [J. Rosický and W. Tholen, Trans. Am. Math. Soc. 355, No. 9, 3611–3623 (2003; Zbl 1030.55015)], the following definition is given.

Definition 4.2. A combinatorial model category \({\mathcal K}\) with the set of generating cofibrations \(I\) is left determined if the class of weak equivalences is \({\mathcal W}_I\).

Theorem 4.3. The model category of flows is left determined.

Section 5 contains concluding remarks. We provide the full text of this section: “The hypothesis that \({\text{ Top}}\) is locally presentable can be removed. Theorem 3.11 and Theorem 4.3 hold by working in any bicomplete cartesian closed full subcategory of the general category of topological spaces containing all CW-complexes. But then, we have to check that all domains and all codomains of the maps of \(I^+_{gl}\) are small relative to \({\text{ cell}}(I^+_{gl})\). This is done in [P. Gaucher, Homology Homotopy Appl. 5, No. 1, 549–599 (2003; Zbl 1069.55008), Section 11] and there is no known way to avoid the use of some difficult topological arguments. However, the model category of flows is left proper but not cellular because of the presence of \(R: \{0,1\}\to \{0\}\) in the generating cofibrations. So outside the framework of locally presentable categories, we have no tools to prove the existence of any homotopical localization and to study the homotopical localization of \({\text{Flow}}\) with respect to the refinement of observation.”

The category \({\text{ Top}}\) denotes a bicomplete locally presentable cartesian closed full subcategory of the category of general topological spaces containing all CW-complexes.

For example, for any topological space \(X\), we denote by \(Glob(X)\) the flow with \({\mathbb P} Glob(X)=X\), \(Glob(X)^0= \{0, 1\}\) and morphisms \(0 \xrightarrow{x} 1\) defined for all \(x\in X\). This induces a functor from \({\text{ Top}}\) to the category \({\text{ Flow}}\).

The paper under review continues the research of its author, published in papers [Theory Appl. Categ. 16, 59–83 (2006; Zbl 1088.55015); Int. J. Math. Math. Sci. 2007, Article ID 87404, 20 p. (2007; Zbl 1149.55009); New York J. Math. 12, 319–348 (2006; Zbl 1109.55010); New York J. Math. 12, 63–95 (2006; Zbl 1110.55011); Appl. Categ. Struct. 13, No. 5–6, 371–388 (2005; Zbl 1115.68103); Homology Homotopy Appl. 7, No. 1, 51–76 (2005; Zbl 1085.55003)].

Section 2 describes Isaev’s approach to constructing model categories. A locally representable category \(\mathcal K\) is considered. According to [J. Rosický, Appl. Categ. Struct. 17, No. 3, 303–316 (2009; Zbl 1175.55013)], combinatorial model categories were introduced as model categories which are locally presentable and cofibrantly generated. The paper under review cites source [A. Joyal, Notes on quasicategories. available at http://preview.tinyurl.com/o6bkdv8, Proposition E.1.10] leading to the statement that for a given set of maps \(I\), there exists at most one combinatorial model category structure on \({\mathcal K}\) such that the set of generating cofibrations is \(I\) and such that all objects are fibrant. The required results from Isaev’s paper [Theory Appl. Categ. 33, 43–66 (2018; Zbl 1393.55012)] on fibrant objects applied to a locally presentable category are collected in Proposition 2.1.

For \(n\geqslant 1\), denote by \(D^n\) the \(n\)-dimensional disk, and by \(S^{n-1}\) the \((n-1)\)-dimensional sphere. Let \(D^0=\{0\}\) and \(S^{-1}= \emptyset\). Denote by \(C\) the inclusion \(\emptyset \to \{0\}\) and by \(R\) the unique map \(\{0, 1\}\to \{0\}\). Let \[ I^{gl}_+= \{Glob(S^{n-1})\subset Glob(D^n)\}| n\geq 0\} \cup\{ C: \emptyset \to \{0\}, R:\{0, 1\}\to \{0\}\} \] be the set of morphisms in the category \({\text{ Flow}}\).

Theorem 3.11. There exists a unique model category structure on \({\text{ Flow}}\) such that \(I^{gl}_+\) is the set of generating cofibrations and such that all objects are fibrant.

Section 4 is devoted to left determinedness of the model category of flows.

For \({\mathcal C}\) a category, a class \(K\subset Mor({\mathcal C})\) is said to satisfy 2-out-of-3 property if for all composable \(f, g\in Mor({\mathcal C})\) we have that if two of the three morphisms \(f, g\) and the composite \(g\circ f\) is in \(K\), then so is the third.

Definition 4.1. Let \(I\) be a set of maps of a locally presentable category \({\mathcal K}\). A class of maps \(\mathcal W\) is a localizer (with respect to \(I\)) or an \(I\)-localizer if \({\mathcal W}\) satisfies:

– Every map satisfying the RLP with respect to the maps of \(I\) belongs to \({\mathcal W}\).

– \({\mathcal W}\) is closed under retract and satisfies the 2-out-of-3 property.

– The class of maps \(cof(I)\cap {\mathcal W}\) is closed under pushout and transfinite composition.

In particular, the class of all maps is an \(I\)-localizer. The class of \(I\)-localizers is closed under arbitrarily large intersection. Therefore there exists a smallest \(I\)-localizer denoted by \({\mathcal W}_I\).

In [J. Rosický and W. Tholen, Trans. Am. Math. Soc. 355, No. 9, 3611–3623 (2003; Zbl 1030.55015)], the following definition is given.

Definition 4.2. A combinatorial model category \({\mathcal K}\) with the set of generating cofibrations \(I\) is left determined if the class of weak equivalences is \({\mathcal W}_I\).

Theorem 4.3. The model category of flows is left determined.

Section 5 contains concluding remarks. We provide the full text of this section: “The hypothesis that \({\text{ Top}}\) is locally presentable can be removed. Theorem 3.11 and Theorem 4.3 hold by working in any bicomplete cartesian closed full subcategory of the general category of topological spaces containing all CW-complexes. But then, we have to check that all domains and all codomains of the maps of \(I^+_{gl}\) are small relative to \({\text{ cell}}(I^+_{gl})\). This is done in [P. Gaucher, Homology Homotopy Appl. 5, No. 1, 549–599 (2003; Zbl 1069.55008), Section 11] and there is no known way to avoid the use of some difficult topological arguments. However, the model category of flows is left proper but not cellular because of the presence of \(R: \{0,1\}\to \{0\}\) in the generating cofibrations. So outside the framework of locally presentable categories, we have no tools to prove the existence of any homotopical localization and to study the homotopical localization of \({\text{Flow}}\) with respect to the refinement of observation.”

Reviewer: Ahmet A. Khusainov (Komsomolsk-om-Amur)

### MSC:

18C35 | Accessible and locally presentable categories |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

68Q85 | Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.) |