diff --git a/thesis/notes/CDGA_Of_Polynomials.tex b/thesis/notes/CDGA_Of_Polynomials.tex index 2d947e9..d1e380b 100644 --- a/thesis/notes/CDGA_Of_Polynomials.tex +++ b/thesis/notes/CDGA_Of_Polynomials.tex @@ -7,7 +7,7 @@ We will now give a cdga model for the $n$-simplex $\Delta^n$. This then allows f where $\deg{x_i} = 0$. So it is the free cdga with $n+1$ generators and their differentials such that $\sum_{i=0}^n x_i = 1$ and in order to be well behaved $\sum_{i=0}^n d x_i = 0$. } -Note that the inclusion $\Lambda(x_1, \ldots, x_n, d x_1, \ldots, d x_n) \to \Apl_n$ is an isomorphism of cdga's. So $\Apl_n$ is free and (algebra) maps from it are determined by their images on $x_i$ for $i = 1, \ldots, n$ (also note that this determines the images for $d x_i$). This fact will be used throughout. +Note that the inclusion $\Lambda(x_1, \ldots, x_n, d x_1, \ldots, d x_n) \to \Apl_n$ is an isomorphism of cdga's. So $\Apl_n$ is free and (algebra) maps from it are determined by their images on $x_i$ for $i = 1, \ldots, n$ (also note that this determines the images for $d x_i$). This fact will be used throughout. Also note that we have already seen the dual unit interval $\Lambda(t, dt)$ which is isomorphic to $\Apl_1$. These cdga's will assemble into a simplicial cdga when we define the face and degeneracy maps as follows ($j = 1, \ldots, n$): diff --git a/thesis/notes/Homotopy_Augmented_CDGA.tex b/thesis/notes/Homotopy_Augmented_CDGA.tex index 4741a53..8ff1498 100644 --- a/thesis/notes/Homotopy_Augmented_CDGA.tex +++ b/thesis/notes/Homotopy_Augmented_CDGA.tex @@ -1,5 +1,5 @@ -Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit} \k$ (this implies that $\counit \unit = \id$). This is precisely the dual notion of a pointed space. We will use the general fact that if $\cat{C}$ is a model category, then the over (resp. under) category $\cat{C} / A$ (resp. $A / \cat{C}$) for any object $A$ admit an induced model structure. In particular, the category of augmented cdga's (with augmentation preserving maps) has a model structure with the fibrations, cofibrations and weak equilavences as above. +Recall that an augmented cdga is a cdga $A$ with an algebra map $A \tot{\counit} \k$ (this implies that $\counit \unit = \id$). This is precisely the dual notion of a pointed space. We will use the general fact that if $\cat{C}$ is a model category, then the over (resp. under) category $\cat{C} / A$ (resp. $A / \cat{C}$) for any object $A$ admit an induced model structure. In particular, the category of augmented cdga's (with augmentation preserving maps) has a model structure with the fibrations, cofibrations and weak equivalences as above. Although the model structure is completely induced, it might still be fruitful to discuss the right notion of a homotopy for augmented cdga's. Consider the following pullback of cdga's: \[ \xymatrix{ diff --git a/thesis/notes/Minimal_Models.tex b/thesis/notes/Minimal_Models.tex index 33dfe2e..58f954e 100644 --- a/thesis/notes/Minimal_Models.tex +++ b/thesis/notes/Minimal_Models.tex @@ -132,7 +132,7 @@ Now if the map $f$ is a weak equivalence, both maps $\phi$ and $\psi$ are surjec \begin{proof} Since both $M$ and $M'$ are minimal, they are cofibrant and so the weak equivalence is a strong homotopy equivalence (\CorollaryRef{cdga_homotopy_properties}). And so the induced map $\pi^n(\phi) : \pi^n(M) \to \pi^n(M')$ is an isomorphism (\LemmaRef{cdga-homotopic-maps-equal-pin}). - Since $M$ (resp. $M'$) is free as a cga's, it is generated by some graded vector space $V$ (resp. $V'$). By an earlier remark \todo{where?} the homotopy groups were eassy to calculate and we conclude that $\phi$ induces an isomorphism from $V$ to $V'$: + Since $M$ (resp. $M'$) is free as a cga's, it is generated by some graded vector space $V$ (resp. $V'$). By an earlier remark \todo{where?} the homotopy groups were easy to calculate and we conclude that $\phi$ induces an isomorphism from $V$ to $V'$: \[ \pi^\ast(\phi) : V \tot{\iso} V'. \] Conclude that $\phi = \Lambda \phi_0$ \todo{why?} is an isomorphism. diff --git a/thesis/notes/Polynomial_Forms.tex b/thesis/notes/Polynomial_Forms.tex index eef1f2d..6bf1bee 100644 --- a/thesis/notes/Polynomial_Forms.tex +++ b/thesis/notes/Polynomial_Forms.tex @@ -42,9 +42,9 @@ where the addition, multiplication and differential are defined pointwise. Concl \subsection{The singular cochain complex} -Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's \todo{Choose: normalized or not?}: -$$ C_n = C^\ast(\Delta^n; \k). $$ -The inclusion maps $d^i : \Delta^n \to \Delta^{n+1}$ and the maps $s^i: \Delta^n \to \Delta^{n-1}$ induce face and degeneracy maps on the dga's $C_n$, turning $C$ into a simplicial dga. Again we can extend this to functors by Kan extensions +Another way to model the $n$-simplex is by the singular cochain complex associated to the topological $n$-simplices. Define the following (non-commutative) dga's: +$$ C_n = C^\ast(\Delta^n; \k), $$ +where $C^\ast(\Delta^n; \k)$ is the (normalized) singular cochain complex of $\Delta^n$ with coefficients in $\k$. The inclusion maps $d^i : \Delta^n \to \Delta^{n+1}$ and the maps $s^i: \Delta^n \to \Delta^{n-1}$ induce face and degeneracy maps on the dga's $C_n$, turning $C$ into a simplicial dga. Again we can extend this to functors by Kan extensions \cdiagram{C_Extension} \todo{show that $C^\ast$ really is sing cohom} where the left adjoint is precisely the functor $C^\ast$ as noted in \cite{felix}. We will relate $\Apl$ and $C$ in order to obtain a natural quasi isomorphism $A(X) \we C^\ast(X)$ for every $X \in \sSet$. Furthermore this map preserves multiplication on the homology algebras.