f468dad6c16c
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Benedikt Fluhr <http://bfluhr.com>
6 years ago

More thorough Treatment of -D-Categories A more thorough (not perfect) treatment of -D-categories.

2 files changed,40insertions(+),20deletions(-) M complPersistenceEnhancements.md M negEnhJoinTrees.md

M complPersistenceEnhancements.md +38 -17

@@ 15,18 15,19 @@ If it wasn't possible for $a$ or $b$ tothen 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 reason why we included infinity is that then we can get an +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 compare the interleaving distance of -the Reeb precosheaf to that of the join tree we have to introduce +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 $\Ec$ but for the monoidal operation -there is some ambiguity, when adding $\infty$ and $-\infty$. +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 $\infty - \infty := -\infty$.@@ 46,13 47,37 @@ to the [category of endofunctors][endofuWe 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 $\mathcal{C}^{\op}$ 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 and let $A$ and $B$ be +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 $B$* +*$(\mathbf{a}, \mathbf{b})$-interleaving of $A$ and $B$ in $\mathbf{C}$* +is an +$(\mathbf{a}, \mathbf{b})$-interleaving of $B$ and $A$ in $\mathbf{C}^{\op}$ +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 $\mathbf{C}^{\op}$ + 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 $\psi \colon \mathcal{S}(-\mathbf{b})(B) \rightarrow A$@@ 86,19 111,15 @@ and$$ commute. - We say *$A$ and $B$ are $(\mathbf{a}, \mathbf{b})$-interleaved* -if there is an $(\mathbf{a}, \mathbf{b})$-interleaving -of $A$ and $B$. - -(@dualDCat) *Remark.* -By passing from $\mathbf{C}$ to $\mathbf{C}^{\op}$ -we can turn any $-D$-category into a $D$-category. -In this sense each result from the previous section -has an analogous statement for $-D$-categories. - -Now let us assume $\mathbf{C}$ is an $\Ec$-category +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 $\Ec$-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

M negEnhJoinTrees.md +2 -3

@@ 43,11 43,10 @@ And this concludes our definition of $\mNow for $(a, b) \in D^{\perp}$, the $(a, b)$-interleavings of two join trees are precisely the $(a, b)$-interleavings with respect to $\mathcal{S}$. -In particular the interleaving distances coincide by corollary @abInterlDist -and remark @dualDCat. +In particular the interleaving distances coincide by corollary @abInterlDist. Admittedly it is a bit of a cheat or kind of trivial, since we specifically defined this subcategory $\mathbf{D}$ -so that this is going to work out. +so that this was going to work out. Next we define the functor $\tilde{\mathcal{E}}$. To this end let $f \colon X \rightarrow \R$ and $g \colon Y \rightarrow \R$