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Wild categories

Definition

A wild category is a 1-dimensional approximation of an infinity-category that is definable in Book HoTT. It consists of

• A type $ob A$ of objects
• For objects $x,y$ a type $A(x,y)$ of morphisms
• For each object $x$ an identity $id_x : A(x,x)$
• For objects $x,y,z$ a composition function $\circ : A(y,z) \times A(x,y) \to A(x,z)$
• For objects $x,y$ and a morphism $f:A(x,y)$, equalities $f \circ id_x = f = id_y \circ f$
• For objects $x,y,z,w$ and morphisms $f:A(x,y)$ and $g:A(y,z)$ and $h:A(z,w)$, an equality $h\circ (g\circ f) = (h\circ g)\circ f$.

If each $A(x,y)$ is a set, then a wild category reduces to a precategory, but in general this condition is not imposed. This means that, for instance, there is a nontrivial pentagon identity for the associativities that does not necessarily commute, and so on. However, even lacking these coherence data, a wild category is sufficient for some purposes.

For example, we can define an initial object in a wild category to be an object $0$ such that $A(0,x)$ is contractible for all $x$. In cases when the wild category “is” actually a coherent higher category, this still gives the right answer, and it is sufficient for applications such as producing induction principles for higher inductive types. See the references for more specific examples.

References

• Paolo Capriotti, Nicolai Kraus, Jakob von Raumer, Path Univalent Spaces of Higher Inductive Categories via Complete Semi-Segal Types in Homotopy Type Theory, arxiv, 2017

• Mike Nicolai Shulman, Kraus, Jakob von Raumer, Impredicative Path Encodings, Spaces Part of 3 Higher Inductive Types in Homotopy Type Theory, blog arxiv post, 2019

• Mike Shulman, Impredicative Encodings, Part 3, blog post, 2018