In the previous section we defined positive persistence enhancements of functors on \({\mathbb{R}}\)-spaces and provided one for \(\mathcal{C}\), thereby finally establishing that the interleaving distances of the Reeb precosheaves provide lower bounds to the correspoding distances on functions. The next aim is to connect this to the interleaving distances of join-trees and to proof theorem 9. Unfortunately there is a bit of a problem with our use of infinity. If we consider the diagrams for interleavings in \(D\)-categories, then there is a smoothing functor on the top right but none on the top left. In the diagrams for interleavings of join trees however there is a shift on the top left but none on the top right. If it wasn’t possible for \(a\) or \(b\) to attain the value \(\infty\), then it wouldn’t matter if we shift on the left or on the right. But if \(a\) or \(b\) is infinity, then the corresponding shift on the right is ill-defined. The purpose of including infinity is that we get an \(\infty\)-interleaving from any other interleaving by monotonicity. So to determine the interleaving distances we could start by considering all \(\infty\)-interleavings and then optimize them with respect to the two weightings. The downside is that in order to harness this framework for comparing the interleaving distance of the Reeb precosheaf to that of the join tree, we have to introduce some more terminology.

Above we defined the monoidal poset \(D\). Now the partial order \(\preceq\) canonically extends to \({\overline{{\mathbb{E}}}}\) but for the monoidal operation there is some ambiguity when adding \(\infty\) and \(-\infty\). But it is still possible if we give up commutativity. More specifically we specify \(-\infty + \infty := \infty\) and \(\infty - \infty := -\infty\). So the last term always dominates. Now \({\overline{{\mathbb{E}}}}\) contains \(D\) as a submonoid. Moreover \(-D\) is a commutative submonoid of \({\overline{{\mathbb{E}}}}\) as well and negation yields an order-reversing monoid isomorphism from \(D\) to \(-D\). Similarly to \(D\)-categories we now define \(-D\)- and \({\overline{{\mathbb{E}}}}\)-categories.

**Definition.**A*strict \({\overline{{\mathbb{E}}}}\)-category*respectively*\(-D\)-category*is a category \(\mathbf{C}\) with a strict monoidal functor \(\mathcal{S}\) from \({\overline{{\mathbb{E}}}}\) respectively \(-D\) to the category of endofunctors on \(\mathbf{C}\). We refer to \(\mathcal{S}\) as the*smoothing functor of \(\mathbf{C}\)*.*Remark.*If \(\mathbf{C}\) is a strict \(-D\)-category with smoothing functor \(\mathcal{S}\), then the opposite category \(\mathbf{C}^{{\operatorname{op}}}\) is a strict \(D\)-category with smoothing functor \(\mathcal{S}(- (\_))\).

Now we define interleavings in \(-D\)-categories. To this end let \(\mathbf{C}\) be a \(-D\)-category with smoothing functor \(\mathcal{S}\) and let \(A\) and \(B\) be objects of \(\mathbf{C}\).

**Definition.**For \((\mathbf{a}, \mathbf{b}) \in \mathcal{D}\) an*\((\mathbf{a}, \mathbf{b})\)-interleaving of \(A\) and \(B\) in \(\mathbf{C}\)*is an \((\mathbf{a}, \mathbf{b})\)-interleaving of \(B\) and \(A\) in \(\mathbf{C}^{{\operatorname{op}}}\) with respect to the smoothing functor \(\mathcal{S}(- (\_))\).We say

*\(A\) and \(B\) are \((\mathbf{a}, \mathbf{b})\)-interleaved*if there is an \((\mathbf{a}, \mathbf{b})\)-interleaving of \(A\) and \(B\).Similarly an

*\((a, b)\)-interleaving of \(A\) and \(B\) in \(\mathbf{C}\)*is an \((a, b)\)-interleaving of \(B\) and \(A\) in \(\mathbf{C}^{{\operatorname{op}}}\) for \((a, b) \in D^{\perp}\).

For convenience we spell out the meaning of the previous definition.

*Remark.*For \((\mathbf{a}, \mathbf{b}) \in \mathcal{D}\) an \((\mathbf{a}, \mathbf{b})\)-interleaving of \(A\) and \(B\) in \(\mathbf{C}\) is a pair of homomorphisms \(\varphi \colon \mathcal{S}(-\mathbf{a})(A) \rightarrow B\) and \(\psi \colon \mathcal{S}(-\mathbf{b})(B) \rightarrow A\) such that the diagrams \[ \xymatrix{ \mathcal{S}(- (\mathbf{a} + \mathbf{b}))(A) \ar@/^/[rr]^{\quad \mathcal{S}(- (\mathbf{a} + \mathbf{b}) \preceq \mathbf{o})_A} \ar[dr]|-{\mathcal{S}(-\mathbf{b})(\varphi)} & & A \\ & \mathcal{S}(-\mathbf{b})(B) \ar[ru]|-{\psi} } \] and \[ \xymatrix{ \mathcal{S}(- (\mathbf{a} + \mathbf{b}))(B) \ar@/^/[rr]^{\quad \mathcal{S}(- (\mathbf{a} + \mathbf{b}) \preceq \mathbf{o})_B} \ar[dr]|-{\mathcal{S}(-\mathbf{a})(\psi)} & & B \\ & \mathcal{S}(-\mathbf{a})(A) \ar[ru]|-{\varphi} } \] commute.

With these definitions we may talk about the absolute and relative
interleaving distance of \(A\) and \(B\). Now let us assume
\(\mathbf{C}\) is a strict \({\overline{{\mathbb{E}}}}\)-category
with smoothing functor \(\mathcal{S}\), then it also is a \(D\)- and
a \(-D\)-category. So if we don’t mention, whether we work with
\(\mathbf{C}\) as a \(D\)-category or as a \(-D\)-category, then our
term *interleaving* is ambiguous. Here (and with
the next lemma) we argue that this creates no problem. Suppose we
have \(\varphi \colon A \rightarrow \mathcal{S}(\mathbf{a})(B)\) for
some \(\mathbf{a} \in D\), then we get the homomorphism
\(\varphi^b := \mathcal{S}(\mathbf{a}-\mathbf{a} \preceq \mathbf{o})_B \circ \mathbf{S}(-\mathbf{a})(\varphi)\)
from \(\mathcal{S}(-\mathbf{a})(A)\) to \(B\). And if we now apply
the functor \(\mathcal{S}(\mathbf{a})\) to \(\varphi^b\) and
precompose with
\(\mathcal{S}(\mathbf{o} \preceq -\mathbf{a}+\mathbf{a})_A\), then
we reobtain \(\varphi\), using that \(\mathcal{S}\) is monoidal.
Such considerations lead us to the following

**Lemma.**The interleavings of \(A\) and \(B\) with respect to the \(D\)-category sructure of \(\mathbf{C}\) are in a canonical bijection with the interleavings of \(A\) and \(B\) with respect to the \(-D\)-category sructure.

Homomorphisms of \(-D\)- and \({\overline{{\mathbb{E}}}}\)-categories are defined completely analogously to those of \(D\)-categories.

**Definition.**Here we define the categories \(-\mathbf{F}\) and \(\pm \mathbf{F}\).The class of objects of \(-\mathbf{F}\) respectively \(\pm \mathbf{F}\) is the class of all pairs \((f, \mathbf{a})\), where \(f \colon X \rightarrow {\mathbb{R}}\) is a continuous real-valued function and \(\mathbf{a} \in -D\) respectively \(\mathbf{a} \in {\overline{{\mathbb{E}}}}\).

Next we specify the homomorphsims of \(\mathbf{F}\). To this end let \(f \colon X \rightarrow {\mathbb{R}}\) and \(g \colon Y \rightarrow {\mathbb{R}}\) be continuous functions and \((a, b), (c, d) \in {\overline{{\mathbb{E}}}}\). Then a

*homomorphism from \((f, (a, b))\) to \((g, (c, d))\)*is a continuous map \(\varphi \colon X \rightarrow Y\) such that \(b - d \leq f(p) - g(\varphi(p)) \leq a - c\) for all \(p \in X\). Whenever there is any ambiguity interpreting the left-hand side of this inequality, we interpret it as \(-\infty\), for the right-hand side as \(\infty\).

Next we define negative and complete persistence-enhancements. To this end let \(F\) be a functor from the category of \({\mathbb{R}}\)-spaces to some category \(\mathbf{C}\).

**Definition.**A*negative*respectively a*complete persistence-enhancement of \(F\)*is the structure of a strict \(-D\)-category respectively \({\overline{{\mathbb{E}}}}\)-category on \(\mathbf{C}\) together with a \(1\)-homomorphism \(\tilde{F}\) from \(-\mathbf{F}\) resepctively \(\pm \mathbf{F}\) to \(\mathbf{C}\) such that \(F = \tilde{F}((\_, \mathbf{o}))\).