While reading these notes I had something of an existential crisis, after realizing that my understanding of direct limits might somehow be fundamentally insufficient. In particular, alarms started going off after reading (9.13), on page 45, wherein the standard result is proven that filtered direct limits of flat modules are themselves flat.

Here's a quick reproduction of the proof in question:

Let $M_i$ be a filtered system of flat modules, and $\beta: N \to N'$ any injective morphism. We have $\varinjlim(M_i \otimes \beta)$ is injective, by the exactness of filtered limits, and since $M_i \otimes \beta$ is injective for each $i$ by flatness. Therefore, since tensor products preserve direct limits, $\varinjlim(M_i\otimes\beta)$=$\varinjlim(M_i)\otimes\beta$, and thus $\varinjlim(M_i)$ is flat.

I have been familiar with this result for some time, and proven it myself in a very similar fashion as above. However, the method of proof employed in these particular notes involve taking limits of morphisms directly ( $\varinjlim(M_i \otimes \beta)$), rather than, as I would usually do something like this, taking kernels, so that relevant morphisms can be constructed after the limits are taken via universal properties. I don't know what it even means to take the limit of a family of morphisms, especially when they are such that there is no common a domain or codomain (so that a comma category construction doesn't suffice). With that in mind I have several closely related questions:

  1. Does there exist a coherent and consistent way of defining the direct limit of a family of morphisms in general?

I assume that this is impossible. Then we interpret $\varinjlim(M_i \otimes \beta)$ as a limit of functors $(\varinjlim(M_i\otimes\bullet))(\beta)$ rather than a limit of morphism per se, which makes everything much more ordinary. However in that case, the latter isomorphism becomes $\varinjlim(M_i\otimes\bullet)$=$\varinjlim(M_i)\otimes\bullet$, which does not follow from immediately from preservation of limits, since the limit is being taken in respect to the functors themselves.

  1. Is there a simple argument, preferably using symmetry and preservation of limits, that yields this identity? We know that these two functors agree on objects (up to natural isomorphism), does that means that they must agree on morphisms too (obviously, this isn't the case in, say, a groupoid)?
  2. Conversely, are there well behaved bifunctors that behave differently in this respect? Something that preserves limits in the product category, but which misbehaves enough that the direct limit of its restriction in one variable is not equal to its restriction to the corresponding direct limit? This probably can't happen in abelian categories but perhaps they exist somewhere else.
  3. If (1) is possible, what is the class of bifunctors such that taking the direct limit of its restrictions and then evaluating at a morphism is the same thing as merely taking the direct limit of its evaluations.
  • $\begingroup$ It's just a bad notation. Actually in the argument they're picking the colimits with the domain and codomain independently and, then, inducing a morphism between these colimits. $\endgroup$ – user40276 Aug 16 '15 at 5:29

Fortunately this has little do to with bifunctors. First of all, note that limits are functorial: given two diagrams $D$, $D' : \mathscr I → \mathscr D$ and a natural transformation $α : D → D'$, we can get a morphism $\lim α : \lim D → \lim D'$. This is easy to construct and write out in some explicit examples of limits (eg. try to do it for products). In particular, taking limits of morphisms very much does make sense. In your case, tensoring with $β$ commutes with everything on the left, which means that $- ⊗ β$ is a natural transformation between $- ⊗ N$ and $- ⊗ N'$ (this is always the case with bifunctors), and then also between diagrams $i ↦ M_i ⊗ N$ and $i ↦ M_i ⊗ N'$.

Now let $G$, $G' : \mathscr D → \mathscr C$ be two limit-preserving functors, and $D : \mathscr I → \mathscr D$ a diagram with a limit in $\mathscr D$. For every natural transformation $β : G ⇒ G'$, $βD$ is a natural transformation $GD ⇒ G'D$ of diagrams in $\mathscr C$, and we have that $\lim βD = β_{\lim D}$ - this follows immediately from the definition of $\lim βD$.

Applied to your example where $- ⊗ N$ and $- ⊗ N'$ are colimit preserving functors, and $- ⊗ β$ is a natural transformation between them, we get that $\varinjlim (M_i ⊗ β) = (\varinjlim M_i) ⊗ β$. There's no need to talk about a limit of functors (but for the record the above is equal to $\varinjlim (M_i ⊗ -)(β)$ too, by the standard result on pointwise limits in functor categories).


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.