## Towards pasting in, and presentations of, semi-strict low dimensional higher categories with weak interchange

### Adrian Miranda – 8 March 2023

This talk is mostly a response to Mike's from last week. I won't have colourful props but we might see a coloured PROP in the sense of [1] as part of a counterexample.

I'll discuss some of the subtleties involved in the combinatorics of n-dimensional paths for small n. For n =1 we get ordinary paths while for n=2 the definition makes well-definedness and uniqueness of a pasting easy to recognise, despite uniqueness often failing to hold. These are 2-cells in free sesquicategories on 2-computads.

When n=3 we will meet the formal interchangers which John mentioned in the discussion after Mike's talk. Just as generating 3-cells in Gray computads are between parallel 2-paths, generating 4-cells in "weak interchange 4-computads" (to be introduced) will be defined between parallel 3-paths. These 3-paths will be built out of generating 3-cells, as well as those formal interchangers. As an example, I'll give a presentation for the free living triadjunction.

[1] https://ncatlab.org/nlab/show/PROP

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