In *Set Theory* Jech defines a cone to be a subset of the Baire Space $\mathcal{N}$ of the form 
$$\operatorname{cone}(x\_0)= \{x : x\_0 \in L[x]\}$$
where $x\_0 \in \mathcal{N}$. Jech then defines the equivalence relation $\equiv$ with 
$$x \equiv y \iff (x \in L[y] \land y \in L[x]).$$
In a similar fashion, in *Set Theory*, Schindler defines a Turing cone to be a subset of the Baire Space of the form 
$$\operatorname{cone}\_T(x\_0) =\{x : x\_0 \leq\_T x \}$$
where $\leq\_T$ denotes Turing reductibility. Schindler then defines the equivalence relation $\equiv\_T$ with 
$$x \equiv\_T y \iff (x \leq\_T y \land y \leq\_T x).$$ 
They both want to prove essentially the same thing, which is:



> 
> Assume $AD$. Then if $A \subset \mathcal{N}$ is a $\equiv$(resp.
>  $\equiv\_T$)-closed subset, then either it or its complement contains a
>  (resp.Turing)cone.
> 
> 
> 


However, what they both do is pick for instance $\sigma$ a winning strategy for $I$ in $G\_A$, and then show that $\operatorname{cone}(\sigma) \subset A$. Now of course, this isn't well defined, as $\sigma \notin \mathcal{N}$ (obviously).
 Now you can of course widen the notion of cones, but then the risk is losing a few characteristics of the filter you're trying to build (as for instance its non principality). 
How do you do this well, to solve this issue and not lose the properties that you require ? 



Defining cones and Turing cones
Simon Thomas already mentioned that a winning strategy can be recursively coded into a real. I thought it might be a good idea to write down one such coding explicitly:


Let $A \subseteq \mathcal{N}$ and let $\Sigma$ be a winning strategy for $G\_{A}$ (say for player $II$, the other case is virtually the same). Then
$$
\Sigma \colon \bigcup\_{n < \omega} \mathbb{N}^{2n+1} \to \mathbb N.
$$
Let $\mathbb P = \{ p\_i \mid i < \omega \}$ be the increasing enumeration of all primes and let
$$
\Sigma^c\colon \mathbb N \to \mathbb N, \prod\_{p \in \mathbb P} p^{n\_p} \mapsto \Sigma(n\_3, \ldots, n\_{p\_{(2 \cdot n\_2 + 1)}}).
$$
Then $\Sigma^c \in \mathcal{N}$ is a recursive coding of $\Sigma$.


