# HG changeset patch # User Benedikt Fluhr # Date 1497843584 -7200 # Mon Jun 19 05:39:44 2017 +0200 # Node ID ba3c80e3b24594a8b9ed3c62424b63a6bd304f54 # Parent f97fa41e608c9f072b92ac15e08ce4c33c71b680 Renamed extended Pers.-Enhancement to complete ... diff --git a/00_09_extPersistenceEnhancements.md b/00_09_complPersistenceEnhancements.md rename from 00_09_extPersistenceEnhancements.md rename to 00_09_complPersistenceEnhancements.md --- a/00_09_extPersistenceEnhancements.md +++ b/00_09_complPersistenceEnhancements.md @@ -1,4 +1,4 @@ -# Extended Persistence Enhancements +# Complete Persistence Enhancements In the previous section we defined positive persistence enhancements of functor on $\R$-spaces and provided one for $\mathcal{C}$, @@ -121,7 +121,6 @@ Homomorphisms of $-D$- and $\Ec$-categories are defined completely analogously to those of $D$-categories. -Now we provide a negative and extended version of $\mathbf{F}$. * **Definition.** Here we define the categories $-\mathbf{F}$ and $\pm \mathbf{F}$. @@ -143,12 +142,12 @@ of this inequality, we interpret it as $-\infty$, for the right-hand side as $\infty$. -Next we define negative and extended persistence-enhancements. +Next we define negative and complete persistence-enhancements. To this end let $F$ be a functor from the category of $\R$-spaces to some category $\mathbf{C}$. * **Definition.** -A *negative* respectively a *extended persistence-enhancement of $F$* +A *negative* respectively a *complete persistence-enhancement of $F$* is the structure of a strict $-D$-category respectively $\Ec$-category on $\mathbf{C}$ together with a $1$-homomorphism $\tilde{F}$ from $-\mathbf{F}$ resepctively $\pm \mathbf{F}$ diff --git a/00_10_someEquivalences.md b/00_10_someEquivalences.md --- a/00_10_someEquivalences.md +++ b/00_10_someEquivalences.md @@ -292,7 +292,7 @@ $\overline{\R}_{-\infty}$ the structure of an $\Ec$-category. So by lemma @pmDEquiv, from the section on -[extended persistence enhancements][], +[complete persistence enhancements][], the interleavings of $\mathcal{C} \mathcal{E} f$ and $\mathcal{C} \mathcal{E} g$ with respect to the $-D$-category structure given by