Most texts on category theory define a (small) diagram in a category $\mathcal{A}$ as a functor $D : \mathcal{I} \to \mathcal{A}$ on a (small) category $\mathcal{I}$, called the shape of the diagram. A cone from $A \in \mathcal{A}$ to $D$ is a morphism of functors $\Delta(A) \to D$, a limit is a universal cone. Observe that, however, that composition in $\mathcal{I}$ is never used to define the limit. One can therefore argue, and this is what I would like to discuss here, that directed multigraphs ("categories without composition") are better suited as the shapes of diagrams:
If $\Gamma$ is a directed multigraph, then a diagram of shape $\Gamma$ in $\mathcal{A}$ is a morphism of graphs $D : \Gamma \to U(\mathcal{A})$, where $U$ forgets composition. A cone from $A \in \mathcal{A}$ to $D$ is a morphism of diagrams $\Delta(A) \to D$, a limit is a universal cone. In my category theory textbook (published 2015) I chose this definition, which leads to an equivalent theory, but offering several advantages over the more common definition:
- As alreay indicated, the limit of a functor $\mathcal{I} \to \mathcal{A}$ in $\mathcal{A}$ is just the limit of the graph morphism $U(\mathcal{I}) \to U(\mathcal{A})$ in $\mathcal{A}$, so it seems awkward to have a category structure around when we do not use it all. Conversely, the limit of a graph morphism $\Gamma \to U(\mathcal{A})$ is just the limit of the corresponding functor $\mathrm{Path}(\Gamma) \to \mathcal{A}$, so in end we end up with the same limits. In particular, the definition cannot be totally wrong, and much of the discussion will be more of philosophical or pedagogical nature.
- When we talk about specific types of diagrams and limits, we never really care about composition, and also never write down identities, since they are not relevant at all. For example, binary products are limits of shape$$\bullet ~~ \bullet$$which is just a graph with two vertices and no edges. We don't need to write down identity morphisms in this approach. Arbitrary products are similar. An equalizer is a limit of the shape
$$\bullet \rightrightarrows \bullet$$
which is just a graph with two vertices and two parallel edges between them. A fiber product is a limit of the shape
$$\bullet \rightarrow \bullet \leftarrow \bullet.$$
Limits of shape
$$\cdots \to \bullet \to \bullet \to \bullet$$
also appear very naturally. Put differently, the typical indexing categories you will find in most texts on category theory are actually already the path categories on directed multigraphs. For me this is the most convincing argument. Barr and Wells argue in their book Toposes, Triples and Theories in a similar way:
Limits were originally taken over directed index sets—partially ordered
sets in which every pair of elements has a lower bound. They were quickly generalized to arbitrary index categories. We have changed this to graphs to reflect actual mathematical practice: index categories are usually defined ad hoc and the composition of arrows is rarely made explicit. It is in fact totally irrelevant and our replacement of index categories by index graphs reflects this fact. There is no gain—or loss—in generality thereby, only an alignment of theory with practice.
- Let's talk about interchanging limits. The usual formulation starts with a functor $D : \mathcal{I} \times \mathcal{J} \to \mathcal{A}$. This includes, in particular all "diagonal" morphisms $D(f,g)$ for morphisms $f$ in $\mathcal{I}$ and $g$ in $\mathcal{J}$. However, in practice, I only want to define $D(f,j)$ and $D(i,g)$, and I don't want to show that $D$ is a functor. For example, interchanging fiber products should be about commuting diagrams of shape
$$\begin{array}{ccccc}
\bullet & \rightarrow & \bullet & \leftarrow & \bullet \\ \downarrow && \downarrow && \downarrow \\ \bullet & \rightarrow & \bullet & \leftarrow & \bullet \\ \uparrow && \uparrow && \uparrow \\ \bullet & \rightarrow & \bullet & \leftarrow & \bullet\end{array}$$
which actually appear in practice (see also here). I don't want to bother about all the diagonal morphisms (and the identities) in that diagram, and actually nobody does when applying "interchanging limits" in concrete examples. The theorem for directed multigraphs is as follows: Let $\Gamma,\Lambda$ be directed multigraphs. Consider the tensor product $\Gamma \otimes \Lambda$ (pair the vertices, pair edges in $\Gamma$ with vertices of $\Lambda$, and pair edges in $\Lambda$ with vertices in $\Gamma$) and a diagram $D$ of shape $\Gamma \otimes \Lambda$ in $\mathcal{A}$ such that for all edges $i \to j$ in $\Gamma$ and edges $i' \to j'$ in $\Lambda$ the diagram
$$\begin{array}{ccc} D(i,j) & \rightarrow & D(i,j') \\ \downarrow && \downarrow \\ D(i',j) & \rightarrow & D(i',j') \end{array}$$
commutes. Then, we have $\lim_{i \in \Gamma} \lim_{j \in \Lambda} D(i,j) \cong \lim_{(i,j) \in \Gamma \otimes \Lambda} D(i,j)$; when the left side exists, then also the right side, and they are isomorphic. - This is a bit vague, but for me it seems awkward and random, almost like a "type error", that categories have two purposes in the usual theory: One purpose it to collect structured objects and their morphisms. The other purpose is to axiomatize diagram shapes. Similarly, functors have two purposes in the usual theory. I find it quite pleasant when the second purpose is fulfilled by a different thing. Also connected to that is the observation that shapes are usually small, but categories tend to be large.
Although the theory works out very well, meanwhile, I am not so confident anymore about my decision, and I am thinking about changing it in the next edition of the book. So here are some disadvantages:
- 99% of the category theory literature (textbooks and research papers) define diagrams as functors, resp. their shapes are just small categories. It is awkward to do something which nobody else does, and this can also be irritating for the readers as well. I didn't bother about this too much when writing the book, but I am increasingly worried about this issue.
- Directed diagrams/colimits are indexed by directed partial orders, and here we really want a functor to ensure compatibility between the various morphisms. Barr-Wells offer a workaround in Chapter 1, Section 10, but they admit themselves that it is slightly awkward.
- The theory of Kan extensions: The left Kan extension of a functor $F : \mathcal{I} \to \mathcal{A}$ along a functor $G : \mathcal{I} \to \mathcal{J}$ at $J \in \mathcal{J}$ can usually be described as the colimit of the functor $G \downarrow J \to \mathcal{I} \to \mathcal{A}$, and it seems artificial to just consider the underlying graph of $G \downarrow J$ here.
This explains hopefully enough background for the following
Questions.
- Can you list further mathematical advantages and disadvantages when taking directed multigraphs as the shapes of diagrams and limits / colimits?
- Can you name pedagogical advantages and disadvantages of this definition?
- Can you list other textbooks on category theory which use this definition? The book Toposes, Triples and Theories by Barr and Wells is an example, see Chapter 1, Section 7. They also define sketches in a "composition-free" way in Chapter 4. Not a book, but Grothendieck also defines diagrams this way in his famous Tohoku paper Sur quelques points d'algèbre homologique, Section 1.6.
- (More general side question) For those of you who already wrote a book or monograph, what other criteria did you choose to decide if a common definition should be changed? And how did you decide in the end?
Best Answer
I think focusing on graphs is not a good idea. We focus on functors for very good reasons. Here are a few:
There are instances where the point of view of graphs is very useful, though. For instance, there is Nori's construction of an abelian category of motives, which relies heavily on singular cohomology seen as a map of graphs; this is documented in this book of Annette Huber and Stefan Müller-Stach, for instance.
If you really want to focus on graphs, then I guess that, in a monograph on category theory, you may write a chapter explaining why we naturally speak the language of graphs when we express ourselves in the lagnuage of category theory. For instance, the category of small category is monadic on the category of graphs. In particular, any category is a colimit of free categories. There are many fundamental exemples of free categories, and is true that, when we write a diagram explicitely, we only write the images of generators, because this is what working on free objects is good for. It is also interesting to see how caterical constructions (such as colimits) are compatible with the presentabions of categories as colimits of free ones. But you will see then that the colimits of interest in this respect are in fact those which are equivalent to their corresponding 2-colimits (i.e. you will start do do homotopy theory where weak equivalences are equivalences of categories). Expressing a given category as $2$-colimits of elementary free categories can be instructive. Cases where we have such a nice inductive procedure of this kind are interesting in practice: this is what is happening with direct Reedy categories, for instance.