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).

(This is a standard term in category theory, but its meaning certainly is not obvious, and I would consider it an error in the book that Riehl uses the term without first defining it.)

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 $y$".

  • $\begingroup$ Couldn't we well order the reals and index objects by that well-ordering? Then an object could have countably-infinite many predecessors? $\endgroup$
    – Aurel
    Sep 25 '18 at 23:26
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    $\begingroup$ 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 $z$". $\endgroup$ Sep 25 '18 at 23:29
  • $\begingroup$ Sorry, I'm confused about this - in the OP's example, there is no morphism f: x -> z, so how is it equivalent to that equivalence relation? $\endgroup$ Jun 1 '20 at 14:55
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    $\begingroup$ @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). $\endgroup$ Jun 1 '20 at 15:27
  • $\begingroup$ Ahh, fantastic - that makes a lot of sense. Thanks! $\endgroup$ Jun 1 '20 at 16:12

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