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I'm currently reading the book "Topoi: The Categorial Analysis of Logic" by Robert Goldblatt, and in chapter 3.11, in order to define limits and co-limits he defines a diagram in a category as follows:


Definition 1:

By a diagram $D$ in a category $\cal C$ we simply mean a collection of $\cal C$-objects $(d_i)_{i\in I}$ together with some $\cal C$-arrows $g:d_i\to d_j$ between certain objects in the diagram.


However, when I looked at the same topic in other books, they all defined a diagram in a category in the following way:


Definition 2: Let $\cal J$ be a (small) category category. A diagram in $\cal C$ is a functor $F:\mathcal{J}\to \mathcal{C}$.


Definition 1 just seems like a "dumbed down" version of definition 2, and maybe the author chose this definition because at this point in the book he didn't even introduce functors.

My question is whether these definitions are equivalent. For example, $F$ being a functor implies that for any $a\in \cal J$ we have $F(id_a)=id_{F(a)}$, so we are considering that in our diagram we have identity morphism associated with every object, and this is not the case using definition 1 of a diagram.

As far as I know, the existence of identity morphisms in the diagram makes no difference when defining the limit and colimit of the diagram, however, there might be definitions in the future that use diagrams as a building block and require them to have identities.

I am asking this question because definition 1 is way more intuitive and easier to visualize, at least for me, and not only that, but it makes understanding cones and limits of diagrams way easier because it simplifies a lot the notation. However, if definition 2 is more useful and standard in the long run I'd rather know that now so I can start using it and building some intuition around it as soon as possible.

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2 Answers 2

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I agree with your "dumbed-down" assessment, and I have to say that I don't rate the scholarship of Goldblatt's book too highly. (Not just my opinion, either: see what Colin McLarty says here.) If you want to learn topos theory from the beginning, I think a better choice would be the book by Moerdijk and Mac Lane, supplemented by whatever "first course in category theory" textbooks you might need to fill in gaps. (I grew up on Mac Lane's book, but there are a lot of newer generation texts that are very good, e.g., the ones by Awodey, Leinster, Riehl, and sorry to leave others out.)

One problem with the dumbed-down definition is that it suppresses some information that you might really want; for example, in addition to "some arrows", maybe it's important whether the composites of some of those arrows are equal to other arrows. The idea of a reflexive coequalizer, for example, is technically important. The way it ought to be expressed is that it's the universal cocone for a functor $D: S \to C$ where the "shape category" $S$ is presented by generating arrows

$$a \overset{i}{\to} b \underset{q}{\overset{p}{\rightrightarrows}} a$$ subject to equations $pi = 1_a = qi$. Goldblatt's explanation would omit this detail. (I guess he could go back and insert this detail by hand after the fact, but I think it'd be better to get it straight from the beginning.)

Also, it's generally a good idea to organize diagrams $D$ according to their underlying shape categories $S$. That's not in his explanation, either. He lumps all diagrams in a category $C$ together, without that refinement. For example, functors may preserve colimits of one shape but not of another, and it's just easier speaking of these things using Definition 2.

So, I won't advise to discard Goldblatt's book altogether: if it gives some explanations that you find more intuitively accessible, then great, but I would definitely cross-check against other books that are closer to "the real deal", written by people who are without doubt practicing category theorists [I mean no disrespect to him, but I don't believe he is one], and spend some time reconciling the points of view as you are doing now.

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  • $\begingroup$ The paper by McLarty that you linked was a very interesting read! $\endgroup$ Feb 23 at 18:05
  • $\begingroup$ A claim in the linked article of McLarty is Grothendieck himself won a Fields Medal in the 1950s for work in functional analysis... $\endgroup$
    – Jochen
    Feb 23 at 19:40
  • $\begingroup$ @Jochen You're quite right; Grothendieck won it in 1966. But it was awarded in large for his work in the 1950's (e.g., homological algebra, K-theory): proofwiki.org/wiki/Definition:Fields_Medal/Recipients/1966. According to some, hsm.stackexchange.com/a/14253/1987, he was already on the list for 1958. $\endgroup$
    – user43208
    Feb 23 at 21:05
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    $\begingroup$ The claim that Grothendieck won the Fields Medal for his work in functional analysis is definitely wrong. $\endgroup$
    – Jochen
    Feb 23 at 21:34
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Definition 1 is more talking about way we would draw a diagram, as a literal picture. For example, consider the "diagram" below $\require{AMScd}$ \begin{CD} B @>{g}>> D\\ @A{f}AA @AA{k}A \\ A @>>{h}> C \end{CD} We left out the arrows $gf$ and $kh$ in the drawing, which should be there according to definition 2. But fine, maybe that will just be a convention of drawing pictures: don't include superfluous information (like identity arrows).

There is another important difference though, consider the simpler triangle diagram consisting of $A \xrightarrow{f} B \xrightarrow{g} C$ and $A \xrightarrow{h} C$ (you'll have to draw the picture yourself now, I do not know how to draw a triangle in MathJax). In definition $2$ there might be the implied condition that $h = gf$, if the category $\mathcal{J}$ is the triangle (due to functoriality of $F$). Note that this may not always be the case, for example we could get the same picture if $\mathcal{J}$ is the category with 3 separate arrows: $X \to X'$, $Y \to Y'$ and $Z \to Z'$. In this case there is no requirement for things to commute. In fact, the latter shows that diagrams in the sense of definition 1 are always diagrams in the sense of definition 2: just consider the category $\mathcal{J}$ with an appropriate number of separate arrows.

In other words: definition 2 can impose conditions on parts of the diagram that need to commute, and definition does not allow for this.

In relation to limits and colimits: when taking the (co)limit of a diagram we do want condition 2. Mainly because it allows us to be more precise about the kind of (co)limits that we want to consider, by talking about a more exact "shape" of their diagram. See also user43208's answer for more details on this.

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  • $\begingroup$ I think you have inverted definition $1$ with definition $2$ in the "in other words" part $\endgroup$
    – Temoi
    Feb 23 at 15:53
  • $\begingroup$ Definition 2 doesn't imply anything about that equation $h = gf$ (nor should anyone expect it to). A useful notion to bear in mind is that of presentation of a category, where the generators are given by a directed graph (aka a quiver), and relations are imposed by equations between arrows in the path category = category freely generated by the quiver; these equations generate a categorical congruence on that free category. So the triangle itself would be the generating quiver, and if no one imposes equations, then the shape category is the free category on the displayed quiver. $\endgroup$
    – user43208
    Feb 23 at 16:25
  • $\begingroup$ @user43208 Yes, you can freely generate, but as the definition is stated there is nothing like that happening. I guess that you are still right in a different sense, namely that $\mathcal{J}$ could just be three separate arrows (e.g. $X \to X'$, $Y \to Y'$ and $Z \to Z'$). I think the main point still stands: in definition 2 one can impose commutativity conditions on diagrams. I'll edit accordingly. $\endgroup$ Feb 23 at 17:12
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    $\begingroup$ @Temoi You are right, I'll edit. $\endgroup$ Feb 23 at 17:12
  • $\begingroup$ If you think the commutativity is forced because you assume that the shape of the diagram is the underlying quiver of a category (i.e., is showing all the arrows of the category), then you've seriously misunderstood, because for example essentially no one draws in identity arrows when they draw a diagram. Another example: think of the p-adics as the limit of a suitable sequence of arrows $\to \bullet \to \ldots \to \bullet$ in the category of rings. It's understood that the arrows of the shape diagram there generate the shape category referred to in definition 2. $\endgroup$
    – user43208
    Feb 23 at 18:33

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