I'm reading a "Graph Transformations. An Introduction to the Categorical Approach" by H.J.Schneider. In a example 6.3.3 Graph Category constructed as a comma category of a identity functor $id_{Set} : Set \rightarrow Set$ and a diagonal functor $\Delta_{Set} : Set \rightarrow Set \times Set$. The functors has a different codomains and couldn't be used to construct a comma category. Does it all mean that a $Set \times Set$ category is a subcategory of $Set$? And that comma category doesn't require equality of the functor codomains? Codomain of the one functor may be a subcategory of a codomain of the another functor?

  • 4
    $\begingroup$ Diagonal functor from exercise 6.3.3 and diagonal functor from nlab are different things. In the example the codomain of diagonal functor is $\mathbf{Set}$, not $\mathbf{Set}\times\mathbf{Set}$. This functor sends every set $X$ to the set $X\times X\in\mathbf{Set}$, not to the pair $(X,X)\in\mathbf{Set}\times\mathbf{Set}$. $\endgroup$ – Oskar Jan 14 '14 at 9:02
  • 1
    $\begingroup$ Thanks! You dispelled my doubts :) But it's strange that a different things has a same name... $\endgroup$ – Denis Jan 14 '14 at 9:11
  • 2
    $\begingroup$ @Denis It's not strange: it's an unavoidable fact that there are only finitely many short words in any language... $\endgroup$ – Zhen Lin Jan 14 '14 at 9:15
  • 2
    $\begingroup$ @Denis I would not call it strange, I would rather say that it is unfortunate and unnecessary for an author to mis-use a well established functor name to represent something much less useful like his X functor. In his defence , at least the author clearly defines what he means and uses the letter X to hint at a cross product. $\endgroup$ – magma Jan 14 '14 at 14:17

Besides the mis-naming (as explained in the comments above), Example 6.3.3 is fine and shows that Graph (the category of directed multi-graphs) can be expressed as a comma category.

You may be interested in knowing that it can also be represented as a functor category:

$$\mathbf{Graph} = \mathbf{Set}^{\Gamma}$$

where $\Gamma$ is the category with exactly two objects and two distinct, parallel, non-identity arrows. In other words: A graph is a functor $G:\Gamma \rightarrow \mathbf{Set}$ and the natural transformations between these functors are the graph transformations. You can read this in Awodey (chap 7 I think).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.