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$\newcommand{\al}{\alpha}$Let $(M_\alpha)_\alpha$ be a direct system of abelian groups, and $\varinjlim M_\alpha$ its direct limit. Then one can show that every element of $\varinjlim M_\alpha$ can be represented by an element $m_\alpha\in M_\alpha$ for some $\alpha$, and using this, it is easy to show that if $J\subset I$ is a cofinal subset, then $\varinjlim_{\al\in J}M_\al\simeq\varinjlim_{\al\in I}M_\al$. I am wondering if one can generalize these two results to a more general setting of category theory. That is, 1) Is there a category theory version of the first result? And 2) Is the second result true in category theory? If so, how can one prove it?

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Do you mean somehow replace the notion of a cofinal subset by some categorical notion? – Asaf Karagila Jan 27 '13 at 13:53
No, let's leave the indexing set as it is. So we are talking about a direct system of objects in a category. – ashpool Jan 27 '13 at 13:56
A nice and rather detailed statement and element-free proof can be found on the last two pages of (found by searching for "cofinal subset direct limit"). – Martin Jan 27 '13 at 13:57
I hope that someday this misleading terminology "direct limit" will be replaced by the correct one: directed colimit. – Martin Brandenburg Jan 27 '13 at 14:15
up vote 4 down vote accepted

2) Colimits for cofinal subcategories are the same. See Mac Lane, Categories for the working mathematician, section IX.3.

1) This is more subtle. What should be an element of an object $X$? In the spirit of the Yoneda Lemma, it makes sense to regard arbitrary morphisms $A \to X$ as "generalized elements" of $X$. Because then the Yoneda Lemma essentially says that an object is determined by its generalized elements, which resembles the extensionality axiom in set theory.

Assume we have a directed colimit $\mathrm{colim}_i X_i$ in a category $C$, and that $A$ is a finitely presentable object of $C$. By definition, this means $\hom(A,\mathrm{colim}_i X_i) = \mathrm{colim}_i \hom(A,X_i)$, so that in fact every generalized element of $\mathrm{colim}_i X_i$ comes from a generalized element from some $X_i$, and that this choice is essentially unique: Two generalized elements of $X_i$ and $X_j$ become equal in the colimit if and only if they become equal in $X_k$ for some $k \geq i,j$. Similar statements hold when the diagram is $\lambda$-directed and $A$ is $\lambda$-presentable for some cardinal $\lambda$.

For example, if $C$ is some algebraic category, then the free object on one generator $F(1)$ of $C$ is finitely presentable (in fact, any object defined by finitely many generators and relations, see Chapter 3 in the book by Adamek and Rosicky on locally presentable categories), which essentially means that the forgetful functor to sets preserves directed colimits. This is not true for arbitrary concrete categories! For example, consider the category of topological vector spaces. The forgetful functor doesn't preserve directed colimits. An element in a colimit of topological vector spaces $V_i$ is just a limit of elements in the $V_i$.

Another example, let $C$ be the category of quasi-coherent sheaves on a concentrated scheme $X$ (for example, noetherian schemes are concentrated). Then $\mathcal{O}_X$ is presentable, which essentially means that the functor of global sections preserves directed colimits. This then can be generalized to its derived functors, i.e. sheaf cohomology, which is quite important and useful.

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The generalization of this simplification of indexing is the notion of a final functor.

A generalization of your first fact is that, in some categories, directed colimits commute with finite limits. In particular, if

$$ M = \varinjlim M_\alpha $$

and I have a subobject $L \mapsto M$, then I also have

$$ L = \varinjlim (M_\alpha \times_M L) $$

expressing $L$ as a colimit of subobjects of the $M_\alpha$'s.

Examples of categories where this is true is Set, AbGrp, any abelian category, any topos, and any variety of universal algebra. (e.g. Rng)

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Yes, any algebraic category has this property. – Martin Brandenburg Jan 27 '13 at 14:06
@MartinBrandenburg what do you mean by algebraic category? Borceux defines an algebraic category as one equivalent to the category of models of some algebraic theory (Vol II, Thm 3.9.1). On the other hand, some people take this term to mean Lawvere theories, i.e categories with finite products where any object is a power of some object $T$. – Exterior Feb 17 '15 at 9:13

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