61b936f5dd92
—
Benedikt Fluhr <http://bfluhr.com>
7 years ago

Reformulations

4 files changed,32insertions(+),29deletions(-) M 00_04_precosheaves.md M 00_05_monoidalPosets.md M 00_06_interleavingReebGraphs.md M 00_07_InterleavingsInDCats.md

M 00_04_precosheaves.md +3 -3

@@ 178,9 178,9 @@ and this yields a natural homomorphism$\eta^{\varphi}_F \colon F \rightarrow \varphi_p \varphi_* F$ for any precosheaf $F$ on $X$. - Conversely we have for all open distinguished open subsets -$V \subseteq Y$ that -$(\varphi^{-1} \circ \varphi^{+1})(V) \subseteq V$. + Conversely we have +$(\varphi^{-1} \circ \varphi^{+1})(V) \subseteq V$ +for all distinguished open subsets $V \subseteq Y$. This induces a natural homomorphism $\varepsilon^{\varphi}_G \colon \varphi_* \varphi_p G \rightarrow G$ for any precosheaf $G$ on $Y$.

M 00_05_monoidalPosets.md +6 -6

@@ 11,11 11,11 @@ The reason for this is two fold.Starting from the approach by @deSilva2016 the author added one layer of formalism to define the relative interleaving distance and -then another layer of formalism -that may help with the computation of these interleaving distances. +then another layer of formalism, +which may help with the computation of these interleaving distances. The nice thing about join trees is that we can get both benefits with just one additional layer of formalism -whereas here we need two layers so that +whereas here we need two layers, so that a dedicated section is required. In other words we beg the reader to bear with us for one more section until we get to the absolute and relative interleaving distances@@ 30,7 30,7 @@ we define a partial order $\preceq$ on $$(x, y) \preceq (x', y') : \Leftrightarrow x \geq x' \wedge y \leq y' .$ -A set augmented with a partial ordering and a monoid structure +A set augmented with a partial ordering and a monotone monoid structure like $D$, we call a *monoidal poset*. Moreover $D \times D$ is a monoidal poset with the product ordering and component-wise addition.@@ 50,7 50,7 @@ We setSo now we have the chain of monoidal posets $\blacktriangledown \subset \triangledown \subset \mathcal{D}$. For each of the two inclusions we aim to describe a -monotone subadditive map into the other direction. +monotone subadditive map in the other direction. * **Definition.** For $(a, b; c, d) \in \mathcal{D}$ we set@@ 76,7 76,7 @@ with $\mathbf{A} \preceq \mathbf{B}$we have $\delta(\mathbf{A}) \preceq \mathbf{B}$. Similarly we have -$\delta(\mathbf{A}) \preceq \mathbf{B}$ +$\gamma(\mathbf{A}) \preceq \mathbf{B}$ for all $\mathbf{A} \in \triangledown$ and $\mathbf{B} \in \blacktriangledown$ with $\mathbf{A} \preceq \mathbf{B}$.

M 00_06_interleavingReebGraphs.md +12 -9

@@ 13,12 13,12 @@ Using the notion of an intersection-basethat @deSilva2016 define the Reeb precosheaf as a precosheaf on the intersection-based space, that has the real numbers as an underlying set and the open intervals as distinguished open subsets. -We will use a different space but all of the distinguished open subsets -of this space except for three can be identified with an open interval. +We will use a different space, but all of the distinguished open subsets +of this space, except for three, can be identified with an open interval. Now because precosheaves are just functors on the distinguished open subsets, this is pretty much the same data. -And the reason that all except for three distinguished open subsets -can be identified with open intervals is that we work with $\overline{\R}$ +And the reason we can't identify all distinguished open subsets +with an interval is that we work with $\overline{\R}$ instead of $\R$, in some sense $\overline{\R}$ has three more open intervals than $\R$. Now we begin with defining this intersection-based space.@@ 30,12 30,12 @@ a precosheaf with further properties.* **Definition.** We define $\overline{\R}_{\infty}$ to be the space that has $\overline{\R}$ as an underlying set and -$\{\overline{\R}\} \cup \{(a, \infty]\}_{a \in \overline{\R}}$ +$\{\overline{\R}\} \cup \{(a, \infty] ~ | ~ a \in \overline{\R}\}$ as it's intersection-base. Similarly we define $\overline{\R}_{-\infty}$ to be the space with the same underlying set and -$\{\overline{\R}\} \cup \{[-\infty, b)\}_{b \in \overline{\R}}$ +$\{\overline{\R}\} \cup \{[-\infty, b) ~ | ~ b \in \overline{\R}\}$ as it's intersection-base. Finally we set@@ 43,11 43,11 @@ as it's intersection-base.$\overline{D} := \set{(x, y)}{\Ec}{x \leq y}$. We note that any functor from the category of topological spaces -to the category of sets defines in particular a precosheaf +to the category of sets defines a precosheaf on any topological space $X$, since we may equip any open subset $U \subseteq X$ with the subspace topology and since inclusions of subsets are continuous with respect to this topology. -In the following definition we define such a functor on the category +In the following definition we define a functor on the category of topological spaces and in this way also a precosheaf on any topological space.@@ 83,10 83,13 @@ we setWith these definitions $\mathcal{C}$ defines a functor from the category of $\overline{\R}$-spaces to the category of set-valued precosheaves on $\overline{D}$. -Now assume that $g$ is a constructible $\overline{\R}$-space. +Now assume that $g$ is a constructible $\overline{\R}$-space[^constr]. We discuss the relationship of the Reeb graph $\mathcal{R} g$ and the Reeb precosheaf $\mathcal{C} g$. +[^constr]: see the [first appendix](#constructible-spaces) +for our definition of a constructible $\overline{\R}$-space + (@CpiIso) **Lemma.** The homomorphism $(\mathcal{C} \circ \pi)_g$ from $\mathcal{C} g$ to $\mathcal{C} \mathcal{R} g$

M 00_07_InterleavingsInDCats.md +11 -11

@@ 212,8 212,8 @@ We will now use this corollary to give mof the interleaving distances. The reason we didn't define the interleaving distances with these more concise descriptions in the first place -is that we believe that having those additional interleavings -around can help with the computation of the interleaving distances. +is that we believe, having those additional interleavings around, +can help with the computation of the interleaving distances. Now suppose $(a, b; c, d) \in \triangledown$, then $(a, b; c, d) = (-d, b; -b, d)$. Thus we have the bijection@@ 242,7 242,7 @@ Let $\mathcal{J}$ be the set of all $(a,* *Proof.* Let $\mathcal{I}'$ be as in the previous corollary. -By the above observations we have $\mathcal{I}' = \Psi(J)$ +By the above observations we have $\mathcal{I}' = \Psi(J)$, thus in conjunction with the previous corollary $\mu_{\mathcal{S}} (A, B) = \inf (\epsilon \circ \Psi)(\mathcal{J})$ and@@ 363,12 363,12 @@ Further we have\mathcal{S}(\mathbf{a} \preceq \mathbf{a} + \mathbf{c} + \mathbf{d})$, since $\mathcal{S}$ is strict monoidal and thus the lower triangle commutes. -For the square to commute we observe that +For the square to commute, we observe that $\mathcal{S}(\mathbf{a} + \mathbf{b} \preceq \mathbf{a} + \mathbf{b} + \mathbf{c} + \mathbf{d}) = \mathcal{S}(\mathbf{a} \preceq \mathbf{a} + \mathbf{c} + \mathbf{d}) \circ - \mathcal{S}(\mathbf{b})$ + \mathcal{S}(\mathbf{b})$, again because $\mathcal{S}$ is strict monoidal. Since all inner triangles and the square commute in the above diagram, the outer triangle commutes as well.@@ 464,7 464,7 @@ We read the second diagram of the above\mathcal{S}' (\mathbf{a} \preceq \mathbf{b}) \circ F$. Now let $F \colon \mathbf{C} \rightarrow \mathbf{C}'$ -be a $1$-homomorphism from and +be a $1$-homomorphism and let $A$ and $B$ be objects of $\mathbf{C}$. <!-- and let $(\mathbf{a}, \mathbf{b}) \in \mathcal{D}$. -->@@ 485,7 485,7 @@ and$\mu_{\mathcal{S}'} (F(A), F(B)) \leq \mu_{\mathcal{S}} (A, B)$. * **Lemma.** -Now suppose that $F$ is [full][] and [faithful][] as a functor +Now suppose that $F$ is [faithful][] as a functor from $\mathbf{C}$ to $\mathbf{C}'$ and that $\varphi \colon A \rightarrow \mathcal{S}(\mathbf{a})(B)$ and@@ 502,7 502,7 @@ Then $\varphi$ and $\psi$ are anThe previous two lemmata have the following * **Corollary.** -If $F$ is full and faithful, then +If $F$ is [full][] and [faithful][], then $F$ induces a bijection between the $(\mathbf{a}, \mathbf{b})$-interleavings of $A$ and $B$ and the $(\mathbf{a}, \mathbf{b})$-interleavings of $F(A)$ and $F(B)$@@ 526,11 526,11 @@ then $G \circ F$ is a $1$-homomorphism fNow we defined $1$-homomorphisms as functors with special properties. For any two functors $F$ and $G$ from $\mathbf{C}$ to $\mathbf{C}'$ we have the class of natural transformations from $F$ to $G$. -Now let $F$ and $G$ also be $1$-homomorphisms, +Now suppose $F$ and $G$ are $1$-homomorphisms, then $F$ and $G$ are in some sense compatible with $\mathcal{S}$ and $\mathcal{S}'$. We name a natural transformation from $F$ to $G$, -that is also compatible with $\mathcal{S}$ and $\mathcal{S}'$, +that is compatible with $\mathcal{S}$ and $\mathcal{S}'$, a *$2$-homomorphism from $F$ to $G$*. * **Definition** (2-Homomorphism) **.**@@ 567,7 567,7 @@ This definition does not include the con\mathcal{S}' (\mathbf{a} \preceq \mathbf{b}) \circ \eta$ for $\mathbf{a} \preceq \mathbf{b}$, where $\circ$ is the [Godement product] of natural transformations. -The reason is that this equation follows from the previous two definitions. +The reason is, that this equation follows from the previous two definitions. If we choose one of the two formulas for the Godement product of $\eta$ and $\mathcal{S} (\mathbf{a} \preceq \mathbf{b})$ and rewrite the term using the previous two definitions,