Proof

Every paracompact manifold admits a Riemannian metric, and for any point in a Riemannian manifold there is a geodesically convex neighborhood (any two points in the neighborhood are connected by a unique geodesic in the neighborhood, one whose length is the distance between the points; see for example the remark after lemma 10.3 (Milnor) page 59). It is immediate that a nonempty intersection of finitely many such geodesically convex neighborhoods is also geodesically convex, hence contractible.

Proof

By (Greene) every paracompact manifold admits a Riemannian metric with positive convexity radius $r_{\mathrm{conv}}\in ℝ$. Choose such a metric and choose an open cover consisting for each point $p\in X$ of the geodesically convex open subset $U_p:=B_p(r_{\mathrm{conv}})$ given by the geodesic $r_{\mathrm{conv}}$-ball at $p$. Since the injectivity radius of any metric is at least $2r_{\mathrm{conv}}$ it follows from the minimality of the geodesics in a geodesically convex region that inside every finite nonempty intersection $U_{p_1}\cap \cdots \cap U_{p_n}$ the geodesic flow around any point $u$ is of radius less than or equal the injectivity radius and is therefore a diffeomorphism onto its image.

Moreover, the preimage of the intersection region under the geometric flow is a star-shaped region in the tangent space $T_{u}X$: because the intersection of geodesically convex regions is itself geodesically convex, so that for any $v\in T_{u}X$ with $\exp (v)\in U_{p_1}\cap \cdots \cap U_{p_n}$ the whole geodesic segment $t\mapsto \exp (tv)$ for $t\in [0,1]$ is also in the region.

So we have that every finite non-empty intersection of the $U_p$ is diffeomorphic to a star-shaped region in a vector space. By the results cited at ball (e.g. theorem 237 of (Ferus)) this star-shaped region is diffeomorphic to an ${ℝ}^n$.

Proof

It is suficcient to check this in $\mathrm{Para}$. We need to check that for $\{U_i\to U\}$ a good open cover and $f:V\to U$ any morphism, we get commuting squares

such that the $\{V_i\to V\}$ form a good open cover of $V$.

Now, while $\mathrm{Para}$ does not have all pullbacks, the pullback of an open cover does exist, and since $f$ is necessarily a continuous function this is an open cover $\{f^{*}{U}_i\to V\}$. The $f^{*}{U}_i$ need not be contractible, but being open subsets of a paracompact manifold, they are themselves paracompact manifolds and hence admit themselves good open covers $\{W_{i,j}\to f^{*}{U}_i\}$.

Then the family of composites $\{W_{i,j}\to f^{*}{U}_i\to V\}$ is clearly a good open cover of $V$.

Proof

It suffices to prove that if $X$ admits a good open cover and $\varphi :S^n\to X$ is an attaching map, then the pushout

also admits a good open cover. Let $\{U_{\alpha }\}$ be a good open cover of $X$ closed under nonempty finite intersections, and choose a contracting homotopy $h_{\alpha }:I\times U_{\alpha }\to U_{\alpha }$ such that $h_{\alpha }(0,-)=\mathrm{id}$ and $h_{\alpha }(1,-)$ is constant. For any subset $S\subseteq D^{n+1}$, let $\mathrm{Hull}(S)$ denote the convex hull of $S$. Then, if $V$ is relatively open in the boundary $S^n$, $\mathrm{Hull}(V)$ is open in $D^{n+1}$. It follows that the image in $Y$ of

is open in $Y$, and it is contractible: define a contracting homotopy

by

These sets $V_{\alpha }$ together with $\mathrm{int}(D^{n+1})$ form a good open cover of $Y$.

Proof

By assumption we have that $C(U)$ is degreewise a coproduct of representables. It is also evidently a split hypercover.

This implies the statement by the characterization of cofibrant objects in the projective structure.