  We introduce the concept of access-based intuitionistic knowledge which
relies on the intuition that agent $i$ knows $\varphi$ if $i$ has found access
to a proof of $\varphi$. Basic principles are distribution and factivity of
knowledge as well as $\square\varphi\rightarrow K_i\varphi$ and
$K_i(\varphi\vee\psi) \rightarrow (K_i\varphi\vee K_i\psi)$, where
$\square\varphi$ reads `$\varphi$ is proved'. The formalization extends a
family of classical modal logics designed in [Lewitzka 2015, 2017, 2019] as
combinations of $IPC$ and $CPC$ and as systems for the reasoning about proof,
i.e. intuitionistic truth. We adopt a formalization of common knowledge from
[Lewitzka 2011] and interpret it here as access-based common knowledge. We
compare our proposal with recent approaches to intuitionistic knowledge
[Artemov and Protopopescu 2016; Lewitzka 2017, 2019] and bring together these
different concepts in a unifying semantic framework based on Heyting algebra
expansions.
