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Contents

Idea

While homotopy type theory formalizes homotopy theory, it is not a priori clear – and in fact is or was an open problem – how to formalize general homotopy-coherent structures of higher algebra/higher category theory: Since these typically involve an infinite hierarchy of coherence-conditions, these cannot be axiomatized directly, but one needs some scheme that generates them. This turned out to be subtle.

Eric Finster had previously considered another variant of type theory, called opetopic type theory which natively talks about infinity-categories and their higher coherences by type-theoretically formalizing the structure of opetopic sets. In new work Finster 18 he gives something like an implementation of aspects of opetopic type theory within homotopy type theory and provides evidence that this is yields a tool to solve the general problem of coherences of higher algebra/higher category theory within homotopy type theory.

References

• Eric Finster (Eric Finster ), ,Towards Higher Universal Algebra in Type Theory, Homotopy Type Theory Electronic Seminar 2018 (pdf slides, recording)

• Agda code at: github.com/ericfinster/higher-alg

• John Baez, James Dolan, Higher-dimensional Algebra and Topological Quantum Field Theory, arXiv:q-alg/9503002

Towards Higher Universal Algebra in Type Theory

Collecting the definitions and trying them out here. The idea is to go through the definitions in the talk and the definitions in the agda formalisation to play around with.

Polynomials

Definition Fix a type $I$ of sorts. A polynomial over $I$, $\Poly I$, is the data of

• A family of operations $\Op : I \to \mathcal{U}$
• For each operation, for each $i : I$, a family of sorted parameters

Remark

• For $i : I$, an element $f: \Op i$ represents an operation whose output sort is $i$.

• For $f : \Op i$ and $j : I$, and element $p : \Param_i(f, j)$ represents an input parameter of sort $j$.

• The $\Op i$ and $\Param_i (f, j)$ are not truncated at set level. So operations and parameters can have higher homotopy.

Trees

A polynomial $P : \Poly I$ generates an assocaiated type of trees.

Definition The inductive family $\Tr P : I \to \mathcal{U}$ has constructors

• $\lf : (i : I) \to \Tr P\, I$
• $\nd : (i : I) \to (f : \Op(P,i)) \to (\phi : (j : J) \to (p : \Param (f, j)) \to \Tr(P, j)) \to \Tr(P, i)$

Leave and Nodes

For a tree $w : \Tr (P, i)$, we will need its type of leaves and type of nodes

Definition

• $\Leaf : (i : I) \to (w : \Tr i) \to I \to \mathcal{U}$
• $\Leaf (\lf i) j := (i = j)$
• $\displaystyle\Leaf (\nd(f, \phi))j := \sum_{k : I} \sum_{p : \Param (f, k)} \Leaf(\phi (k, p)) j$