Paradigm - categories as systems

A 1-category C can be viewed as data

State spaces:

• $C(pt):=Fun(pt,C)$, set of states on a point
• $\{C(M):=Fun(M,C)|M\text{ a 1-manifold with boundary }\}$ set of states on 1-manifolds.

With structures:

• maps restricting a state on a 1-manifold to states on the boundary $C(M)\to C(\partial M)\simeq C(pt{)}^{\bigsqcup (\#\partial M)}$
• maps encoding how to glue states along shared boundaries $C(M){\times }_{\partial M=\partial N}C(N)\to C(M{\cup }_{{\partial }_M}N)$ The gluing data needs to be compatible/associative.

By a cut-and-paste argument, it turns out we really need to $C(M)$ for a single case, namely $M$ being the closed interval $[0,1]$, with boundary $\partial M=\{0,1\}$.

Let’s move up along the categorification ladder. A 2-category can be viewed as data

State spaces:

• C(pt), C(M) for 1-manifolds with boundaries M, C(V) for 2-manifolds with corners V. Structures
• Restrictions to boundaries, for both 2-manifolds and 1-manifolds (in particular, to corners)
• Data gluing 2-manifolds along boundaries and corners

Now by cutting-and-pasting along triangulations of surfaces, we find that we only need to specify $C({\Delta }^0),C({\Delta }^1),C({\Delta }^2)$. where ${\Delta }^1$ is viewed as manifold with boundary, and ${\Delta }^2$ is viewed as a manifold with corners - these are the structures naturally present on the simplexes viewed as subsets of Euclidean space.

To bring homotopy theory into the