diff --git a/thesis/3_SimplicialAbelianGroups.tex b/thesis/3_SimplicialAbelianGroups.tex index c092aad..24bc90e 100644 --- a/thesis/3_SimplicialAbelianGroups.tex +++ b/thesis/3_SimplicialAbelianGroups.tex @@ -39,12 +39,12 @@ This category $\DELTA$ will act as a protoype for these kind of geometric struct (Or equivalently a covariant functor $X: \DELTA^{op} \to \Set.$) \end{definition} -So the category of all simplicial sets, $\sSet$, is the functor category $\Set^{\DELTA^{op}}$, where morphisms are natural transformations. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $X: \DELTA^{op} \to \Set$. And hence we can picture a simplicial set as done in figure~\ref{fig:simplicial_abelian_group}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $X$ is a contravariant functor. +So the category of all simplicial sets, $\sSet$, is the functor category $\Set^{\DELTA^{op}}$, where morphisms are natural transformations. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $X: \DELTA^{op} \to \Set$. And hence we can picture a simplicial set as done in figure~\ref{fig:simplicial_set}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $X$ is a contravariant functor. \begin{figure} - \includegraphics{simplicial_abelian_group} - \caption{A simplicial abelian group.} - \label{fig:simplicial_abelian_group} + \includegraphics{simplicial_set} + \caption{A simplicial set.} + \label{fig:simplicial_set} \end{figure} diff --git a/thesis/images/delta_cat.pdf b/thesis/images/delta_cat.pdf index e363585..41f07f9 100644 Binary files a/thesis/images/delta_cat.pdf and b/thesis/images/delta_cat.pdf differ diff --git a/thesis/images/delta_cat.svg b/thesis/images/delta_cat.svg index 58e72ca..f9dd865 100644 --- a/thesis/images/delta_cat.svg +++ b/thesis/images/delta_cat.svg @@ -13,7 +13,7 @@ version="1.1" id="svg4102" inkscape:version="0.48.2 r9819" - sodipodi:docname="thesissymbols.svg"> + sodipodi:docname="delta_cat.svg"> @@ -22,7 +22,7 @@ image/svg+xml - + @@ -47,7 +47,7 @@ inkscape:cx="117.00119" inkscape:cy="49.609396" inkscape:window-x="0" - inkscape:window-y="0" + inkscape:window-y="626" inkscape:window-maximized="0" inkscape:current-layer="svg4102" /> - - - - diff --git a/thesis/images/simplicial_abelian_group.pdf b/thesis/images/simplicial_abelian_group.pdf deleted file mode 100644 index 5c020bc..0000000 Binary files a/thesis/images/simplicial_abelian_group.pdf and /dev/null differ diff --git a/thesis/images/simplicial_set.pdf b/thesis/images/simplicial_set.pdf new file mode 100644 index 0000000..31d4c57 Binary files /dev/null and b/thesis/images/simplicial_set.pdf differ diff --git a/thesis/images/simplicial_abelian_group.svg b/thesis/images/simplicial_set.svg similarity index 89% rename from thesis/images/simplicial_abelian_group.svg rename to thesis/images/simplicial_set.svg index 76140c5..c4749e4 100644 --- a/thesis/images/simplicial_abelian_group.svg +++ b/thesis/images/simplicial_set.svg @@ -13,7 +13,7 @@ version="1.1" id="svg4102" inkscape:version="0.48.2 r9819" - sodipodi:docname="simplicial_abelian_group.svg"> + sodipodi:docname="simplicial_set.svg"> @@ -22,7 +22,7 @@ image/svg+xml - + @@ -43,11 +43,11 @@ fit-margin-top="1" fit-margin-right="1" fit-margin-bottom="1" - inkscape:zoom="2.1649824" - inkscape:cx="117.00119" - inkscape:cy="49.609396" + inkscape:zoom="4.0400329" + inkscape:cx="128.75472" + inkscape:cy="33.512817" inkscape:window-x="0" - inkscape:window-y="0" + inkscape:window-y="626" inkscape:window-maximized="0" inkscape:current-layer="svg4102" /> - - - - @@ -1212,59 +1198,99 @@ inkscape:connector-curvature="0" /> - - + transform="translate(-241.14743,-102.94531)" + id="g3329"> + + + + + + - - + transform="translate(-210.11118,-102.99289)" + id="g3335"> + + + + + + - - + transform="translate(-179.7478,-102.93373)" + id="g3341"> + + + + + + - - + transform="translate(-149.00333,-103.25566)" + id="g3347"> + + + + + +