Rectification of interleavings and a persistent Whitehead theorem

We work model-independently whenever possible. This means that whenever we say space we will mean either topological space or simplicial set. Results stated for spaces will hold for both possible models. The category of spaces will be denoted by $\SS$.

For a general introduction to simplicial sets see, e.g., [9] or [12, Chapter 3]. We denote the geometric realization functor for simplicial sets by $|-|:\mathbf{sSet}\to\mathbf{Top}$.

The theory of model categories was introduced in [18]; for a modern and thorough development of this theory we recommend [12] and [11].

We recall that two objects $x,y\in\mathcal{M}$ of a model category $\mathcal{M}$ are said to be weakly equivalent if they are isomorphic in $\mathsf{Ho}(\mathcal{M})$, which happens if and only if they are connected by a zig-zag of weak equivalences in $\mathcal{M}$. This is an equivalence relation, which we denote by $x\simeq y$. When there is risk of confusion, morphisms in $\mathsf{Ho}(\mathcal{M})$ will be surrounded by square brackets $[f]$, to distinguish them from morphisms in $\mathcal{M}$.

Two of the main model structures of interest to us are the Quillen model structure on $\mathbf{Top}$, the category of topological spaces ([12, Chapter 1, Section 2.4]), and the Kan–Quillen model structure on $\mathbf{sSet}$, the category of simplicial sets ([12, Chapter 3]). We recall that the geometric realization functor $|-|:\mathbf{sSet}\to\mathbf{Top}$ is left adjoint to the singular functor $\operatorname{Sing}:\mathbf{Top}\to\mathbf{sSet}$, and that, together, they form a Quillen equivalence ([12, Chapter 1, Section 1.3], [12, Theorem 3.6.7]). For completeness, we mention that there is a subcategory $\mathbf{Top}_{\mathsf{CGWH}}\subseteq\mathbf{Top}$, the category of compactly-generated weakly Hausdorff topological spaces (called compactly generated spaces in [12, Definition 2.4.21]), that is often used instead of $\mathbf{Top}$. The Quillen model structure on $\mathbf{Top}$ restricts to a model structure on $\mathbf{Top}_{\mathsf{CGWH}}$, and the inclusion $\mathbf{Top}_{\mathsf{CGWH}}\to\mathbf{Top}$ is part of a Quillen equivalence ([12, Theorem 2.4.25]). This model structure is, in some respects, better behaved than the Quillen model structure on topological spaces, and is in fact the model of space used in [2]. We will not concern ourselves with these subtleties since, by the observations in 2.1, there is no essential difference between using $\mathbf{Top}$ or $\mathbf{Top}_{\mathsf{CGWH}}$ when studying homotopy-invariant notions of interleaving.

We will make use of the notion of cofibrantly generated model category ([12, Chapter 2, Section 2.1]). Recall that the Kan–Quillen model structure on simplicial sets is cofibrantly generated, where a set of generating cofibrations consists of the boundary inclusions $\partial\Delta^{n}\hookrightarrow\Delta^{n}$ for $n\geq 0$ ([12, Theorem 3.6.5]). The Quillen model structure on topological spaces is also cofibrantly generated, with a set of generating cofibrations given by $\{S^{n-1}\hookrightarrow D^{n}\}_{n\geq 0}$ ([12, Theorem 2.4.19]).

We conclude by recalling the basic properties of projective model structures. Given a model category $\mathcal{M}$ and a small category ${\mathbf{I}}$, the projective model structure on the functor category $\mathcal{M}^{{\mathbf{I}}}$ is, when it exists, the model structure whose fibrations (respectively weak equivalences) are those which are pointwise fibrations (respectively weak equivalences) of $\mathcal{M}$.

The projective model structure on $\mathcal{M}^{\mathbf{I}}$ exists, and is cofibrantly generated, whenever $\mathcal{M}$ is cofibrantly generated. Moreover, if ${\mathcal{I}}$ and ${\mathcal{J}}$ are, respectively, generating cofibrations and generating trivial cofibration for the model structure of $\mathcal{M}$, then $\left\{{\mathbf{I}}(i,-)\odot f\ :\ i\in{\mathbf{I}},\,f\in{\mathcal{I}}\right\}$ and $\left\{{\mathbf{I}}(i,-)\odot g\ :\ i\in{\mathbf{I}},\,g\in{\mathcal{J}}\right\}$ are, respectively, generating cofibrations and generating trivial cofibrations for the projective model structure, where, given a functor $F:{\mathbf{I}}\to\mathbf{Set}$ and an object $X\in\mathcal{M}$, the functor $F\odot X:{\mathbf{I}}\to\mathcal{M}$ is defined by $i\mapsto\coprod_{a\in F(i)}X$ ([11, Section 11.6]). For simplicity, we denote ${\mathbf{I}}(i,-)\odot X$ by $i\odot X$.

We are especially interested in the projective model structure when the indexing category is a poset $(P,\leq)$. In this case, if $r\in P$ and $X\in\mathcal{M}$, then $r\odot X$ is the functor that takes the value $X$ on every $s\geq r$, and has as value the initial object of $\mathcal{M}$ when $s\nleq r$. The non-trivial structure morphisms of this functor are the identity of $X$.

Note that we have a functor $\mathsf{h}:\mathsf{Ho}\left(\mathcal{M}^{\mathbf{I}}\right)\to\mathsf{Ho}(\mathcal{M})^{\mathbf{I}}$ by the universal property of $\mathsf{Ho}\left(\mathcal{M}^{\mathbf{I}}\right)$.

We denote the poset of real numbers with their standard order by $\mathbf{R}$, and for $m\in\mathbb{N}$, we let $\mathbf{R}^{m}$ be the set of $m$-tuples of real numbers with the product order. We set $\overline{m}=\{i:1\leq i\leq m\}$, so that $(\varepsilon_{i})_{i\in\overline{m}}\leq(\delta_{i})_{i\in\overline{m}}\in\mathbf{R}^{m}$ if and only if $\varepsilon_{i}\leq\delta_{i}$ for all $1\leq i\leq m$. We denote the element $(0,\dots,0)\in\mathbf{R}^{m}$ by $0$.

Fix a category $C$ and a natural number $m\geq 1$. An $m$-persistent object of $C$ is any functor of the form $\mathbf{R}^{m}\to C$. We often refer to $m$-persistent objects simply as persistent objects or as multi-persistent objects when we want to stress the fact that $m$ is not necessarily $1$. Fix persistent objects $X,Y,Z\in C^{\mathbf{R}^{m}}$, $r,s\in\mathbf{R}^{m}$, and $\varepsilon,\delta\geq 0\in\mathbf{R}^{m}$. We use the following conventions.

• •

  For $f:X\to Y$ a natural transformation, denote the $r$-component of $f$ by $f_{r}:X(r)\to Y(r)$.

• •

  Assume $r\leq s$. The structure morphism $X(r)\to X(s)$ will be denoted by $\varphi^{X}_{r,s}$.

• •

  The $\delta$-shift to the left of $X$ is the functor $X^{\delta}:\mathbf{R}^{m}\to C$ defined by $X^{\delta}(r)=X(r+\delta)$, with structure morphisms $\varphi^{X^{\delta}}_{r,s}:=\varphi^{X}_{r+\delta,s+\delta}$. Shifting to the left gives a functor $(-)^{\delta}:C^{\mathbf{R}^{m}}\to C^{\mathbf{R}^{m}}$. Dually, there is a $\delta$-shift to the right functor $\delta\cdot(-):C^{\mathbf{R}^{m}}\to C^{\mathbf{R}^{m}}$ defined by mapping $X$ to the persistent object $\delta\cdot X$, with values given by $(\delta\cdot X)(r)=X(r-\delta)$.

• •

  Natural transformations $f:X\to Y^{\delta}$ will be referred to as $\delta$-morphisms, and will often be denoted by $f:X\to_{\delta}Y$. Since we have natural bijections $\operatorname{Hom}(\varepsilon\cdot X,Y^{\delta})\cong\operatorname{Hom}(X,Y^{\varepsilon+\delta})\cong\operatorname{Hom}((\varepsilon+\delta)\cdot X,Y)$, we can treat a $\delta$-morphism $f:X\to_{\delta}Y$ as $f:X\to Y^{\delta}$ or as $f:\delta\cdot X\to Y$.

• •

  Assume $\varepsilon\leq\delta$ and let $f:X\to_{\varepsilon}Y$. We can compose the $r$-component of $f$ with $\varphi^{Y}_{r+\varepsilon,r+\delta}:Y(r+\varepsilon)\to Y(r+\delta)$, giving $\varphi^{Y}_{r+\varepsilon,r+\delta}\circ f_{r}:X(r)\to Y(r+\delta)$. Together, these components define the shift from $\varepsilon$ to $\delta$ of $f$, which is a $\delta$-morphism denoted $\mathsf{S}_{\varepsilon,\delta}(f):X\to_{\delta}Y$.

• •

  Note that an $\varepsilon$-morphism $f:X\to_{\varepsilon}Y$ can be composed with a $\delta$-morphism $g:Y\to_{\delta}Z$, yielding an $(\varepsilon+\delta)$-morphism $g^{\varepsilon}\circ f:X\to_{\varepsilon+\delta}Y$. This composition is associative and unital, and is natural with respect to shifts of morphisms.

• •

  An $(\varepsilon,\delta)$-interleaving between $X$ and $Y$ consists of an $\varepsilon$-morphism $f:X\to_{\varepsilon}Y$ together with a $\delta$-morphism $g:Y\to_{\delta}X$ such that $g^{\varepsilon}\circ f=\mathsf{S}_{0,\varepsilon+\delta}(\mathsf{id}_{X})$ and $f^{\delta}\circ g=\mathsf{S}_{0,\varepsilon+\delta}(\mathsf{id}_{Y})$. By $\delta$-interleaving we mean a $(\delta,\delta)$-interleaving.

• •

  If $f:X\to_{\varepsilon}Y$ and $g:Y\to_{\delta}X$ form an $(\varepsilon,\delta)$-interleaving, we write $f:X\prescript{}{{\scriptscriptstyle\delta}}{\longleftrightarrow}_{{\scriptscriptstyle\varepsilon}}\,Y:g$.

Let $\varepsilon_{1},\varepsilon_{2},\delta_{1},\delta_{2}\geq 0\in\mathbf{R}^{m}$. Note that an $(\varepsilon_{1},\varepsilon_{2})$-interleaving between $X$ and $Y$ can be composed with any $(\delta_{1},\delta_{2})$-interleaving between $Y$ and $Z$, yielding an $(\varepsilon_{1}+\delta_{1},\varepsilon_{2}+\delta_{2})$-interleaving. The fact that interleavings compose implies that, when $m=1$, the formula

$$d_{I}(X,Y)=\inf\left(\{\delta\geq 0\in\mathbf{R}:\text{$X$ and $Y$ are $\delta$-interleaved}\}\cup\{\infty\}\right)$$

defines an extended pseudo metric $d_{I}:\mathsf{Obj}\left(C^{\mathbf{R}}\right)\times\mathsf{Obj}\left(C^{\mathbf{R}}\right)\to[0,\infty]$. This is the interleaving distance on the class of objects of the category $C^{\mathbf{R}}$. This notion of distance can be extended to objects of the functor category $C^{\mathbf{R}^{m}}$ ([15]), but we will not make use of this extension.

If one is comparing objects of a category of functors of the form $\mathbf{R}^{m}\to\mathcal{M}$, for $\mathcal{M}$ a model category, it makes sense to want to find a homotopy-invariant notion of interleaving. In this paper, we consider the following three homotopy-invariant relaxations of the notion of interleaving. Let $\mathcal{M}$ be a cofibrantly generated model category and endow $\mathcal{M}^{\mathbf{R}^{m}}$ with the projective model structure. Let $X,Y\in\mathcal{M}^{\mathbf{R}^{m}}$ and let $\varepsilon,\delta\geq 0\in\mathbf{R}^{m}$.

1. 1.

   Following [2], we say that $X$ and $Y$ are $(\varepsilon,\delta)$-homotopy interleaved if there exist $X\simeq X^{\prime}$ and $Y\simeq Y^{\prime}$ such that $X^{\prime}$ and $Y^{\prime}$ are $(\varepsilon,\delta)$-interleaved.

2. 2.

   Note that the shift functor $(-)^{\delta}:\mathcal{M}^{\mathbf{R}^{m}}\to\mathcal{M}^{\mathbf{R}^{m}}$ maps weak equivalences to weak equivalences. This implies that all the notions in Section 2.2.1 have analogues in the category $\mathsf{Ho}\left(\mathcal{M}^{\mathbf{R}^{m}}\right)$. We say that $X$ and $Y$ are $(\varepsilon,\delta)$-interleaved in the homotopy category if they are $(\varepsilon,\delta)$-interleaved as objects of $\mathsf{Ho}\left(\mathcal{M}^{\mathbf{R}^{m}}\right)$.

3. 3.

   Finally, as also done in [2], we say that $X$ and $Y$ are $(\varepsilon,\delta)$-homotopy commutative interleaved if their images $\mathsf{h}X,\mathsf{h}Y:\mathbf{R}^{m}\to\mathsf{Ho}(\mathcal{M})$ are $(\varepsilon,\delta)$-interleaved.

An $(\varepsilon,\delta)$-homotopy interleaving gives rise to an $(\varepsilon,\delta)$-interleaving in the homotopy category, which, in turn, gives rise to an $(\varepsilon,\delta)$-homotopy commutative interleaving.

For each of the three homotopy-invariant notions of interleaving introduced above, we have a corresponding extended pseudo metric on the collection of objects of the category $\mathcal{M}^{\mathbf{R}}$. Let $X,Y\in\mathcal{M}^{\mathbf{R}}$. Following [2], we define the homotopy interleaving distance as

$$d_{HI}(X,Y)=\inf\left(\{\delta\geq 0\in\mathbf{R}:\text{$X$ and $Y$ are $\delta$-homotopy interleaved}\}\cup\{\infty\}\right).$$

The fact that the homotopy interleaving distance satisfies the triangle inequality follows from 2.3. The interleaving distance in the homotopy category is

$$d_{IHC}(X,Y)=\inf\left(\{\delta\geq 0\in\mathbf{R}:\text{$X$ and $Y$ are $\delta$-interleaved in the homotopy category}\}\cup\{\infty\}\right).$$

Again following [2], the homotopy commutative interleaving distance is defined as

$$d_{HC}(X,Y)=\inf\left(\{\delta\geq 0\in\mathbf{R}:\text{$X$ and $Y$ are $\delta$-homotopy commutative interleaved}\}\cup\{\infty\}\right).$$

In this short section, we give a simplified proof of a generalization of the fact that homotopy interleavings can be composed, originally proved in [2, Section 4]. This is generalized further in [20, Theorem 4.1.4].

We remark that the idea of using pullbacks to prove a triangle inequality appears in [17].