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 set
So 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-base
that @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 set
With 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 m
of 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 an
The 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 f
Now 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,