Infinitary logic considers languages being infinite by infinite conjunctions and disjunctions.

I wonder why it not considers languages being infinite by relations and functions of infinite arity.

Relations of finite arity $n$ over a base set $A$ can be seen as unary predicates of functions $f:[n] \rightarrow A$. Nothing prohibits us to consider more general functions $f:\mathbb{N} \rightarrow A$ or even $f:\mathbb{R}^+_0 \rightarrow A$.

Is there a model theory assuming a language that allows for relations and functions of infinite and even uncountable arity?

^{I asked this question at MSE but did get no feedback.}

Higher operads, higher categoriesat arxiv.org/abs/math.CT/0305049 . – Qiaochu Yuan Feb 22 '11 at 21:20