Homotopy Type Theory
Eric Finster, Towards Higher Universal Algebra in Type Theory (Rev #2)

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.

Introduction

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 II of sorts. A polynomial over II, PolyI\Poly I, is the data of

  • A family of operations Op:I𝒰\Op : I \to \mathcal{U}
  • For each operation, for each i:Ii : I, a family of sorted parameters
Param i:(f:Opi)I𝒰\Param_i : (f : \Op i) \to I \to \mathcal{U}

Remark

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

  • For f:Opif : \Op i and j:Ij : I, and element p:Param ifjp : \Param_i f\, j represents an input parameter of sort jj.

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

Trees

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

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

  • lf:(i:I)TrPI\lf : (i : I) \to \Tr P\, I
  • nd:(i:I)(f:OpPi)(ϕ:(j:J)(p:Paramfj)TrPj)TrPi\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:TrPiw : \Tr P\, i, we will need its type of leaves and type of nodes

Definition

  • Leaf:(i:I)(w:Tri)I𝒰\Leaf : (i : I) \to (w : \Tr i) \to I \to \mathcal{U}
  • Leaf(lfi)j:=(i=j)\Leaf (\lf i) j := (i = j)
  • Leaf(nd(f,ϕ))j:= k:I p:ParamfkLeaf(ϕkp)j\displaystyle\Leaf (\nd(f, \phi))j := \sum_{k : I} \sum_{p : \Param f\, k} \Leaf(\phi k\, p) j

Revision on December 7, 2018 at 02:16:12 by Urs Schreiber. See the history of this page for a list of all contributions to it.