Is there a good reference for the study of locally presentable and accessible categories without the axiom of choice? For instance, it seems one will need to understand:
What is a good notion of $\kappa$-small set in ZF?
Do the usual definitions of $\kappa$-filtered colimit and $\kappa$-presentable object require modification?
Can you still prove that a category $\mathcal K$ is of the form $\mathcal K = Ind_\kappa(\mathcal C)$ for $\mathcal C$ a small category iff $\mathcal K$ is locally small and has a small subcategory of $\kappa$-presentable objects of which every object is $\kappa$-filtered colimit?
Can you still prove that $\mathcal K = Ind_\kappa(\mathcal C)$ is complete iff it is cocomplete iff (the idempotent completion of) $\mathcal C$ has $\kappa$-small colimits? Call such a $\mathcal K$ locally $\kappa$-presentable.
Can you still prove the special adjoint functor theorem? I.e. that a functor between locally presentable categories is a left adjoint iff it is cocontinuous and a right adjoint iff it is accessible and continuous?
--insert favorite property of accessible / locally presentable categories here --
Do you need to use the word "anafunctor" to do these things properly?
