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changeset 16:f3c974521ba0
Change of Terminology: D-category instead of D-module
We use the term D-category where we previously used the term D-module
to avoid a clash of terminology,
since D-modules already exist and are something completely different.
This has been pointed out by Michael Catanzaro.
author | Benedikt Fluhr <http://bfluhr.com> |
---|---|
date | Tue, 09 May 2017 17:20:58 +0200 |
parents | 1cb388196827 |
children | 114ba7e6a7f6 |
files | poster.tex |
diffstat | 1 files changed, 7 insertions(+), 7 deletions(-) [+] |
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--- a/poster.tex Mon Apr 17 20:37:35 2017 +0200 +++ b/poster.tex Tue May 09 17:20:58 2017 +0200 @@ -198,15 +198,15 @@ for all \((a, b; c, d) \in \mathcal{D}\). } \column{0.5} - \block{Interleavings in \(D\)-modules}{ - \begin{definition}[\(D\)-modules] - A \emph{\(D\)-module} is a category \(\mathcal{C}\) + \block{Interleavings in \(D\)-categories}{ + \begin{definition}[\(D\)-categories] + A \emph{\(D\)-category} is a category \(\mathcal{C}\) with a strict monoidal functor \(\mathcal{S}\) from \(D\) to the category of endofunctors on \(\mathcal{C}\). We refer to \(\mathcal{S}\) as the \emph{smoothing functor of \(\mathcal{C}\)}. \end{definition} - Now let \(\mathcal{C}\) be a \(D\)-module + Now let \(\mathcal{C}\) be a \(D\)-category with smoothing functor \(\mathcal{S}\). For \(a \leq 0\), \(b \geq 0\), and an object \(A\) in \(\mathcal{C}\) we get two things, @@ -273,10 +273,10 @@ and \(\mu(A, B)\) the \emph{relative interleaving distance}. \end{definition} - Now let \(\mathcal{C}'\) be another \(D\)-module with smoothing + Now let \(\mathcal{C}'\) be another \(D\)-category with smoothing functor \(\mathcal{S}'\). \begin{definition} - A \(1\)-homomorphism of \(D\)-modules + A \(1\)-homomorphism of \(D\)-categories from \(\mathcal{C}\) to \(\mathcal{C}'\) is a functor from \(\mathcal{C}\) to \(\mathcal{C}'\) such that \[ @@ -337,7 +337,7 @@ to a category \(\mathcal{C}\). \begin{definition} A \emph{persistence-enhancement of \(F\)} - is the structure of a \(D\)-module on \(\mathcal{C}\) + is the structure of a \(D\)-category on \(\mathcal{C}\) together with a \(1\)-homomorphism \(\tilde{F}\) from \(\mathcal{F}\) to \(\mathcal{C}\) such that \(\tilde{F}((\_, \mathbf{o})) = F\).