nLab
presentation of a category by generators and relations

Idea

Similarly to the presentation of groups by generators and relations, a category may be presented by a set of generating arrows subject to certain relations.

Definition

Let $G$ be a directed graph, and let $R$ be a function that assigns to each pair $a,b$ of objects of the free category $F(G)$ a binary relation $R_{a,b}$ on the hom-set $F(G)(a,b)$. The category with generators $G$ and relations $R$ is the quotient category (as defined in Mac Lane or Mitchell, for example – this is not the nLab definition due to issues of evil) $F(G)/R$. For a category $C$, an isomorphism $C\to F(G)/R$ is called a presentation of $C$.

Properties

Writing $\mathrm{can}:F(G)\to F(G)/R$ for the canonical functor, it follows from the universal property of the quotient category that for any functor $S:F(G)\to D$ that respects the relation $R$ ($f{R}_{a,b}g$ implies $S(f)=S(g)$), there exists a unique functor $S\prime :F(G)/R\to D$ with $S=S\prime \circ \mathrm{can}$.

Examples

1. Every category $C$ has a presentation by generators and relations: Take $G$ as the underlying graph of $C$, and for objects $a$, $b$, let $R_{a,b}$ be the relation on $F(G)(a,b)$ consisting of all pairs of paths from $a$ to $b$ in $G$ whose arrows have the same composition in $C$. However, there are sometimes more economical presentations for a category, as the following example shows.

2. The augmented simplex category ${\Delta }_a$ is generated by the face maps and the degeneracy maps, subject to the simplicial relations (see simplex category for details). The existence of a functor from the quotient category to ${\Delta }_a$ follows from the fact that the arrows of ${\Delta }_a$ do satisfy the simplicial relations, and the fact that this functor is an isomorphism may be verified using the unique decomposition of an arrow of ${\Delta }_a$ as the composition of degeneracies of decreasing index followed by the composition of face maps of increasing index (see the lemma on p. 177 of Mac Lane). Similarly, the subcategory $({\Delta }_a{)}_{\mathrm{inj}}$ consisting of all monics (injective monotone functions in our case) is generated by the face maps subject to the single simplicial relation involving only face maps.

2-cells

A useful way to think of the relations is as being 2-cells between parallel pairs of arrows, thus if $a,b$ are objects, and $(u,v)\in R_{a,b}$, we think of $(u,v)$ as a 2-cell (initially from $u$ to $v$). In this way, one can encode rewriting systems of a certain kind in terms of the embryonic data for a 2-category. This is discussed more in the entry on computad, which are also called polygraphs.

References

• Saunders Mac Lane, Categories for the Working Mathematician, pp. 51-52

• Barry Mitchell, Introduction to Category Theory and Homological Algebra, in P. Salmon (Ed.), Categories and Commutative Algebra. Springer, 2010. pp. 108-112.

• Barry Mitchell, Rings with several objects, Advances in Mathematics 8 (1972), 1–161.