f468dad6c16c — 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, 40 insertions(+), 20 deletions(-)

M complPersistenceEnhancements.md
M negEnhJoinTrees.md
M complPersistenceEnhancements.md +38 -17
@@ 15,18 15,19 @@ If it wasn't possible for $a$ or $b$ to 
 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 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][endofu
 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 $\mathcal{C}^{\op}$ 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 $\m
 Now 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 $g \colon Y \rightarrow \R$