d8d0d68546ae — Benedikt Fluhr <http://bfluhr.com> 6 years ago
Discussion of the Join Precosheaf
6 files changed, 42 insertions(+), 8 deletions(-)

M EqualityOfInterlDist.md
M complPersistenceEnhancements.md
M posPersistenceEnhancements.md
M someEquivalences.md
M struct.yaml

M EqualityOfInterlDist.md +1 -1
@@ 65,7 65,7 @@ transformations of the smoothing functor
To this end let $\mathbf{a}, \mathbf{b} \in -D$ with
$\mathbf{a} \preceq \mathbf{b}$.

-* **Lemma.**
+(@shiftDescJoinTree) **Lemma.**
We have
$(\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_f =  M complPersistenceEnhancements.md +1 -1 @@ 49,7 49,7 @@ We refer to$\mathcal{S}$as the * *Remark.* If$\mathbf{C}$is a strict$-D$-category with smoothing functor$\mathcal{S}$, -then the opposite category$\mathcal{C}^{\op}$is a strict +then the opposite category$\mathbf{C}^{\op}$is a strict$D$-category with smoothing functor$\mathcal{S}(- (\_))$. Now we define interleavings in$-D$-categories.  M links.md +1 -0 @@ 26,3 26,4 @@ [Godement product]: https://ncatlab.org/nlab/show/Godement+product [strict 2-category]: https://ncatlab.org/nlab/show/strict+2-category [galois]: https://en.wikipedia.org/wiki/Galois_connection#.28Monotone.29_Galois_connection +[category of arrows]: https://unapologetic.wordpress.com/2007/05/23/arrow-categories/  M posPersistenceEnhancements.md +33 -4 @@ 108,12 108,12 @@ to the category of set-valued precosheav such that$\mathcal{C} = \tilde{\mathcal{C}}((\_, \mathbf{o}))$. Applying the same procedure to an arbitrary functor$F$on the category of$\R$-spaces -is what we name a *positive persistence-enhancement of$F$*. +is what we name a *positive persistence-enhancement for$F$*. Now let$F$be a functor from the category of$\R$-spaces to some category$\mathbf{C}$* **Definition** (Positive Persistence-Enhancement)**.** -A *postive persistence-enhancement of$F$* +A *postive persistence-enhancement for$F$* is the structure of a strict$D$-category on$\mathbf{C}$together with a$1$-homomorphism$\tilde{F}$from$\mathbf{F}$to$\mathbf{C}$such that$F = \tilde{F}((\_, \mathbf{o}))$. @@ 131,7 131,7 @@ see [the section on homomorphisms in$D$and the previous corollary. * *Example.* -To conclude this section we provide a positive persistence-enhancements for +As a first example we provide a positive persistence-enhancements for the Reeb precosheaf$\mathcal{C}$. In the previous section we already defined the structure of a strict$D$-category on the category of precosheaves. @@ 141,7 141,7 @@ To this end we set$\Delta^{\mathbf{a}} \colon \R \rightarrow \Ec,
t \mapsto (t, t) - \mathbf{a}$and -$\overline{C}((f, \mathbf{a})) :=
+$\overline{\mathcal{C}}((f, \mathbf{a})) := (\Delta^{\mathbf{a}} \circ f)_* \Lambda$ for any $\mathbf{a} \in D$.
Now let
$\varphi \colon (f, \mathbf{a}) \rightarrow (g, \mathbf{b})$

@@ 229,3 229,32 @@ defines a $1$-homomorphism of strict $D$
$\R$-space and $r \in \R$
we set
-->
+
+Now let $F$ and $G$ be functors from the category of $\R$-spaces
+to some $D$-category $\mathbf{C}$ with smoothing functor $\mathcal{S}$
+and let $\eta \colon F \rightarrow G$
+be a natural transformation.
+We consider $\eta$ a functor from the category of $\R$-spaces
+to the [category of arrows][] in $\mathbf{C}$.
+Moreover we consider the [category of arrows][] in $\mathbf{C}$
+a $D$-category with smoothing functor $\mathcal{S}$.
+(We just apply the smoothing functor to the homomorphisms.)
+Then a (positive) persistence-enhancement of
+$\eta$ with smoothing functor $\mathcal{S}$
+is already determined by the corresponding enhancements for $F$ and $G$.
+Now suppose $\tilde{F}$ and $\tilde{G}$ are arbitrary persistence-enhancements
+for $F$ and $G$ both with smoothing functor $\mathcal{S}$.
+
+* **Definition.**
+We say *$\mathcal{S}$, $\tilde{F}$, and $\tilde{G}$
+combine to a persistence-enhancement of $\eta$*
+if the map
+$(f, \mathbf{a}) \mapsto (\mathcal{S}(\mathbf{a}) \circ \eta)_f$
+is a natural transformation from $\tilde{F}$ to $\tilde{G}$.
+
+(@combine2Hom) *Remark.*
+If $\mathcal{S}$, $\tilde{F}$, and $\tilde{G}$
+combine to a persistence-enhancement of $\eta$,
+then
+$(f, \mathbf{a}) \mapsto (\mathcal{S}(\mathbf{a}) \circ \eta)_f$
+is a $2$-homomorphism from $\tilde{F}$ to $\tilde{G}$.


M someEquivalences.md +2 -2
@@ 142,7 142,7 @@ and this implies the claim.

Now let $g \colon Y \rightarrow \R$ be another continuous function.

-* **Corollary.**
+(@etaCEIsoCor) **Corollary.**
The interleavings of
$\mathcal{C} \mathcal{E} f$ and $\mathcal{C} \mathcal{E} g$
are in bijection to those of

@@ 295,7 295,7 @@ the interleavings of
with respect to the $-D$-category structure given by
$\overline{\mathcal{S}}$, are in canonical bijection with those
given by the $\overline{\mathcal{S}}$-induced structure of a $D$-category.
-So in conjunction with the last corollary from the previous subsection
+So in conjunction with corollary @etaCEIsoCor from the previous subsection
we have the following

(@bijD) **Propostion.**


M struct.yaml +4 -0
@@ 41,6 41,10 @@ sections:
- title: Equality of Interleaving Distances
files:
- EqualityOfInterlDist.md
+- title: Relation to the Join Precosheaf
+  name: join-precosheaf
+  files:
+  - joinPrecosheaf.md
- title: Appendix
sections:
- title: Constructible Spaces over the Reals