# Why is finiteness necessary in definition of connected category

I am reading through Category Theory in Context by Emily Riehl. On page 33 she defines a connected category as one in which any pair of objects can be connected by a finite zigzag of morphisms (emphasis added).

Why do we need to specify a finite zigzag? Given an infinite family of morphisms from an object $$x$$ to an object $$y,$$ couldn't I just compose all of them to get a single morphism from $$x$$ to $$y?$$

My only guess is that "zig-zag" allows for morphisms pointing in opposite directions, e.g.

$$x \rightarrow y \leftarrow z$$

would constitute a zigzag of morphisms. In this case $$x$$ would be connected to $$z$$ (even if $$\text{Hom}(x,z) = \text{Hom}(z,x) = \emptyset$$). On the other hand in the case of a countable family $$\{y_i\}_{i=1}^\infty$$ of objects, the morphisms

$$x \rightarrow y_1 \leftarrow y_2 \rightarrow \cdots \leftarrow z,$$ would not connect $$x$$ and $$z$$.

Is my interpretation correct or am I missing something?

Your guess is correct: the term "zig-zag" refers to a sequence of morphisms which could go in either direction. So, a zig-zag from $$x$$ to $$y$$ is a sequence of objects $$z_0=x,z_1,z_2,\dots,z_n=y$$ together with either a morphism $$z_i\to z_{i+1}$$ or a morphism $$z_{i+1}\to z_i$$ for each $$i$$ from $$0$$ to $$n-1$$. The point of this definition is that "$$x$$ is connected to $$z$$" (i.e., "there exists a zig-zag from $$x$$ to $$z$$") should be the equivalence relation generated by "there exists a morphism from $$x$$ to $$z$$". If you imagine the category as a graph where the objects are vertices and the morphisms are edges, a zig-zag is a path from $$x$$ to $$z$$ (where we don't care about the directions of the arrows).
There is no such thing as an "infinite zig-zag" from $$x$$ to $$y$$ and it would not make sense to talk about such a thing. You can define an infinite zig-zag where your objects are indexed by $$\mathbb{N}$$ (or $$\mathbb{Z}$$), but in that case there would be no last (or first) object and so it would not make sense to say that the zig-zag is "from $$x$$ to $$y$$".
• Well, sure, but that is highly unnatural, since there would be no relationship at all between $z_\omega$ and $z_n$ for finite $n$, for instance. The point of this definition is that "$x$ is connected to $z$" should be the equivalence relation generated by "there exists a morphism from $x$ to $z$". Sep 25 '18 at 23:29
• @It'sNotALie.: Existence of a morphism $x\to z$ is not an equivalence relation--it's typically not symmetric. The equivalence relation I'm talking about is the equivalence relation it generates (in other words, the smallest equivalence relation containing it). Jun 1 '20 at 15:27