# HG changeset patch # User Benedikt Fluhr # Date 1503318248 -7200 # Mon Aug 21 14:24:08 2017 +0200 # Node ID d8d0d68546ae37de5189622cff1ba10c6397f326 # Parent 343ef1eebcbee2293ae7872fdce0643ae1fb2882 Discussion of the Join Precosheaf diff --git a/EqualityOfInterlDist.md b/EqualityOfInterlDist.md --- a/EqualityOfInterlDist.md +++ b/EqualityOfInterlDist.md @@ -65,7 +65,7 @@ To this end let $\mathbf{a}, \mathbf{b} \in -D$ with $\mathbf{a} \preceq \mathbf{b}$. -* **Lemma.** +(@shiftDescJoinTree) **Lemma.** We have $(\overline{\mathcal{S}}(\mathbf{a} \preceq \mathbf{b}) \circ \pi^2_* \circ \mathcal{C} \circ \mathcal{E})_f = diff --git a/complPersistenceEnhancements.md b/complPersistenceEnhancements.md --- a/complPersistenceEnhancements.md +++ b/complPersistenceEnhancements.md @@ -49,7 +49,7 @@ * *Remark.* If$\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$-categories. diff --git a/links.md b/links.md --- a/links.md +++ b/links.md @@ -26,3 +26,4 @@ [Godement product]: https://ncatlab.org/nlab/show/Godement+product [strict 2-category]: https://ncatlab.org/nlab/show/strict+2-category [galois]: https://en.wikipedia.org/wiki/Galois_connection#.28Monotone.29_Galois_connection +[category of arrows]: https://unapologetic.wordpress.com/2007/05/23/arrow-categories/ diff --git a/posPersistenceEnhancements.md b/posPersistenceEnhancements.md --- a/posPersistenceEnhancements.md +++ b/posPersistenceEnhancements.md @@ -108,12 +108,12 @@ such that$\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 @@ 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 @@$\Delta^{\mathbf{a}} \colon \R \rightarrow \Ec, t \mapsto (t, t) - \mathbf{a}$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 @@ $\R$-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 $\tilde{G}$. diff --git a/someEquivalences.md b/someEquivalences.md --- a/someEquivalences.md +++ b/someEquivalences.md @@ -142,7 +142,7 @@ 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 @@ 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.** diff --git a/struct.yaml b/struct.yaml --- a/struct.yaml +++ b/struct.yaml @@ -41,6 +41,10 @@ - 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