large cardinal tree properties as properties of sheaves - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T07:21:47Z http://mathoverflow.net/feeds/question/109695 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109695/large-cardinal-tree-properties-as-properties-of-sheaves large cardinal tree properties as properties of sheaves o a 2012-10-15T08:25:07Z 2012-10-15T22:39:26Z <p>As follows from this talk Large Properties for Small Cardinals, p.7,p.4 <a href="http://www2.dm.unito.it/paginepersonali/viale/SEMINARS-TORINO/Fontanella-Torino-19.1.2012.pdf" rel="nofollow">http://www2.dm.unito.it/paginepersonali/viale/SEMINARS-TORINO/Fontanella-Torino-19.1.2012.pdf</a>, the definitions of weakly compact and strongly compact large cardinals can be formulated stated in the language of sheaf theory. <em>I am wondering whether sheaf theory can be of any help, and whether supercompactness can be reformulated in a similar way.</em> </p> <p>Let me describe these reformulations. </p> <p>For an inaccesible cardinal $\kappa$, a $\kappa$-tree (in set theory) is a <em>sheaf of 0-1 valued functions</em> on ordinal $\kappa$ as a topological space with the initial interval topology: a subset of $\kappa$ is open iff it is an initial segment $\alpha\leq\kappa$. A $\kappa$ tree has <em>a cofinal branch</em> iff the sheaf has <em>a global section</em>, i.e. $T(\kappa)$ is non-empty. For an arbitrary cardinal $\kappa$, a $\kappa$-tree $T(\kappa)$ corresponds to a sheaf $F:\kappa^{op}\longrightarrow Sets$ of 0-1 valued functions such that for every <em>successor</em> ordinal $\beta+1&lt;\kappa$, $|F(\beta+1)|&lt;\kappa$. (Corrections thanks to Joel and Nate's comments). </p> <p>(WC) A inaccessible cardinal $\kappa$ is <em>weakly compact</em> iff any such sheaf of functions of $\kappa$ such that there is a section for every proper open subset $\alpha&lt;\kappa$, has a global section, i.e. eqv., any $\kappa$-tree has a cofinal branch. </p> <p>Given a cardinal $\kappa$, define a Grothendieck topology on (the partial order viewed as) the category $(P(\lambda), \subseteq)$ (where $P(\lambda)=$ {$X: X\subseteq \lambda$} as follows: a collection $S=${$U_i\subseteq U$}$_i$ forms a <em>covering</em> of $U$ iff for every $Y\subseteq U$ of size $|Y|&lt;\kappa$, there exists $U_i\subseteq U$ in $S$ such that $U_i\supseteq Y$. (By definition, a Grothendieck topology is a collection of these coverings; we shall denote $P(\lambda), \subseteq)_\kappa$ this category equipped with this topology.)</p> <p>(SC) An <em>inaccessible</em> cardinal $\kappa$ is <em>strongly compact</em> iff forevery sheaf $T:(P(\lambda), \subseteq)_\kappa \longrightarrow Sets$ of Sets (with that topology), the following implication holds: if $0&lt;|T(X)|&lt;\kappa$ for every $|X|&lt;\kappa$, then $T(\lambda)\neq \emptyset$, i.e. the sheaf has a global section. (Being somewhat vague, one may drop the assumption of inaccessibility and require instead that every sheaf of <em>functions</em> has a global section). Set-theoretically, this is said as any $(\kappa,\lambda)$-tree has a cofinal branch. </p> <p>If I understand correctly, an interesting question is whether for every(?) $\kappa$ every sheaf of sets with the condition $|X|&lt;\kappa$ implies $|T(X)|&lt;\kappa$ is consistent. </p> <p>Let me say more about the definitions of sheaves and how to pass from a tree to a sheaf. Recall (as e.g. stated in the talk above) a $\kappa$-tree $T(\kappa)$ can be viewed as a subset of $2^{&lt;\kappa}$. Define a sheaf $F$ on $\kappa$ as follows: (i) for limit ordinal $\alpha$ $F(\alpha)$ is the set of all functions $f:\alpha\longrightarrow 2$ such that for every $\beta&lt;\alpha$, the restriction of $f$ to $f|\beta:\beta \longrightarrow 2$ is in $T(\kappa)$. (ii) for a successor ordinal, $F(\alpha+1)=Lev_{\alpha+1}(T)$ where $Lev_{\alpha+1}(T)$ is the set $2^{\alpha+1}\cap T(\kappa)$ of functions $f:\alpha+1\longrightarrow 2$ of tree $T(\kappa)$.</p> <p>The condition (*) is the sheaf condition for this topology: $\alpha=\cup{\beta&lt;\alpha}\beta$ is an <em>open covering</em> of open set $\alpha$, and (*) says that every function on the open set $\alpha$ is uniquely defined by its restrictions on the sets in an open covering (and vice versa, any compatible set of functions on $\beta$'s defines a functions on $\alpha$). Note that this condition is only non-trivial for <em>limit</em> ordinals: for successors there is no non-trivial covering. </p> <p>Finally, let me say why (WC) implies that $\kappa$ is inaccessible. Let $\kappa=2^\alpha$ and $\alpha&lt;\kappa$. Let $f_i:\alpha\longrightarrow 2, i&lt;\kappa$ be an enumeration of $2^\alpha$. Define $F(j+1)=${$g: g|\alpha\neq f_i \forall i\leq j$} for every ordinal $j&lt;\kappa$, and at limit stage define $F$ by sheaf property $(*)$ above. Then $F(j)$ is non-empty for every $j&lt;\kappa$, yet $F(\kappa)$ is empty. </p> <p>PS: I thank Joel and Nate for many corrections. </p>