Colimit of a sequence of objects in a category

In the book "Lecture notes in algebraic topology" by Davis and Kirk I came across the following colimit $$E^{\infty} = \operatorname{colim}_{r \to \infty}E^r$$

where each of the $$E^r$$ were $$R$$-modules and $$E^{\infty}$$ was the $$R$$-module defined to be the colimit of the $$E^r$$'s

Now from what I understand about colimits, colimits aren't defined for a sequence of objects in some category (as the above notation seems to suggest), they are defined instead for functors. In the above, it seems that the colimit in question is a sequential colimit.

Now I'm assuming that this collection of $$R$$-modules gives rise to some functor $$T$$ from $$(\mathbb{N}, \leq)$$ (which is the filtered index category on the directed set $$(\mathbb{N}, \leq)$$) to $$\operatorname{R-Mod}$$, for which $$T(n) = E^n$$ and for which $$T$$ acts somehow on morphisms. Then I'm assuming that it will turn out that $$\operatorname{co}\varinjlim T = E^{\infty}$$ and we'd just label $$\operatorname{colim}_{r \to \infty}E^r$$ as $$\operatorname{co}\varinjlim T$$. Am I correct in saying this? If so let me move onto my next question.

The problem is that I don't know actually how to define such a functor $$T$$ on morphisms. The only possible morphisms on the $$E^r$$'s in the above case are endomorphisms $$d^r : E^r \to E^r$$.

Let me now restate this next question to be as self-contained as possible. Suppose that I have a collection of $$R$$-modules, $$E^r$$ for each $$r \in \mathbb{N}$$. How can I define a functor $$T$$ from $$(\mathbb{N}, \leq)$$ to $$\operatorname{R-Mod}$$ such that $$\operatorname{co}\varinjlim T$$ is $$\operatorname{colim}_{r \to \infty}E^r$$?

To give some context because I'm sure my question possibly isn't well-formed, the colimit $$E^{\infty} = \operatorname{colim}_{r \to \infty}E^r$$ is the colimit of the pages/sheets of a spectral seqeunce. See below for an excerpt of the passage where the colimit appears

• In general, it's not true that the $E^\infty$ page of a spectral sequence can be considered as a colimit of the finite pages (as you observe, there aren't actually natural maps $E^r\to E^{r+1}$ in general). Can you give more specific context? Jul 2, 2019 at 16:50
• @EricWofsey Well the textbook I'm working from is Lecture notes in Algebraic Topology by Kirk and Davis (which is available from Davis' site here: indiana.edu/~jfdavis/teaching/m623/book.pdf), the colimit in my question is shown on page 241, I will add an image in my question to show this Jul 2, 2019 at 17:44
• @EricWofsey I just took a look at definition 9.5 (convergence of a homology spectral sequence) on page 243 (after the passage with the colimit in question) and it turns out that convergence implies the existence of a surjection $E^r_{p, q} \to E^{r+1}_{p, q}$ for $r$ greater than some $r_0$ (also note here that the $E^r$ are assumed to be bigraded $R$-modules). I guess these would then be the morphisms Jul 2, 2019 at 17:53

That notion is the (directed) colimit of the diagram $$E_0 \longrightarrow E_1 \longrightarrow E_2 \longrightarrow E_3 \longrightarrow \dots,$$ for some morphisms $$\varphi_{ij} \colon E_i \rightarrow E_j$$ such that $$\varphi_{ik} = \varphi_{jk} \circ \varphi_{ij}$$ for all $$i \leq j \leq k$$ and $$\varphi_{ii} \colon E_i \rightarrow E_i$$ is the identity. That is an object $$E$$ together with morphisms $$\psi_i \colon E_i \rightarrow E$$ satisfying $$\psi_i = \psi_j \circ \varphi_{ij}$$ whenever $$i \leq j$$ which is universal in the following sense:
For each obejct $$F$$ together with morphisms $$\eta_i \colon E_i \rightarrow F$$ satisfying $$\eta_i = \eta_j \circ \varphi_{ij}$$ whenever $$i \leq j$$ there exists exactly one morphism $$f \colon E \rightarrow F$$ such that the diagram
commutes for all $$i,j$$. The notion which is most often used is $$E = \varinjlim E_i$$, where the morphisms etc. are dropped.
In the case of modules the object $$E$$ is realized as a quotient of the direct sum of the $$E_i$$ and can be thought of as "gluing" the $$E_i$$ together according to the given maps.
We can also regard $$\mathbb{N}$$ as a directed index category and then the functor $$\mathcal{F} \colon \mathbb{N} \rightarrow R-\text{mod}$$ defining the $$E_i$$ and the $$\varphi_{ij}$$ as you mentioned.
As Eric stated as well it is not clear what the morphisms between the $$E_i$$ are supposed to be in your situation (at least for me).