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Free nc function theory is an extension of the theory of holomorphic functions of several complex variables to the theory of functions on matrix tuples $Z=(Z_1,\cdots,Z_d)$ where $Z_i\in M_n(\mathbb{C})$ and $n$ is allowed to vary.

An nc function is a function defined on such tuples $Z$ and takes values in $\cup_{n\in \mathbb{N}} M_n(\mathbb{C})$ which is graded and respects direct sums and similarity (equivalently, respects intertwiners).

The classical correspondence between positive kernels and Hilbert spaces of functions has been recently extended by Ball, Marx and Vinnikov to nc completely positive kernels and Hilbert spaces of nc functions. In a previous, unpulished work, we have developed a similar theory for matricial functions where $\mathbb{C}$ is replaced by a von Neumann algebra $M$, $\cup_{n\in \mathbb{N}} M_n(\mathbb{C})$ is replaced by a suitable disjoint union of correspondences over $M$ and the ``index set” $\mathbb{N}$ is replaced by the set of representations of $M$.

Thus, in both situations we deal with structures that are fibred. In each settings there are situations where we need to move among fibres. This led us to consider actions of categories on fibred sets and we study here functions and kernels that are invariant under certain actions of categories.

This is a joint work (in progress) with Paul Muhly.