The theory and applications of anticolimits

TL;DR

It is established that in the presence of pullbacks, there is a "canonical"anticolimit which characterises the existence of other anticolimits, which provides convenient techniques for computing anticolimits, by changing either the shape or ambient category.

Abstract

Colimits are a fundamental construction in category theory. They provide a way to construct new objects by gluing together existing objects that are related in some way. We introduce a complementary notion of anticolimits, which provide a way to decompose an object into a colimit of other objects. While anticolimits are not unique in general, we establish that in the presence of pullbacks, there is a "canonical" anticolimit which characterises the existence of other anticolimits. We also provide convenient techniques for computing anticolimits, by changing either the shape or ambient category. The main motivation for this work is the development of a new method, known as anticontraction, for constructing homotopies in the proof assistant homotopy.io for finitely presented $n$-categories. Anticontraction complements the existing contraction method and facilitates the construction of homotopies increasing the complexity of a term, enhancing the usability of the proof assistant. For example, it simplifies the naturality move and third Reidemeister move.

Figures (8)

• Figure 1: Naturality of the braiding without anticontraction.
• Figure 2: Naturality of the braiding with anticontraction.
• Figure 3: Reidemeister III without anticontraction.
• Figure 4: Reidemeister III with anticontraction.
• Figure 5: Correspondence between hypergraphs and posets.

Theorems & Definitions (72)

• Definition 1
• Example 2
• Proposition 3
• proof
• Definition 4
• Definition 5
• Lemma 6
• proof
• Lemma 7
• proof