Explicit description of colimits of algebras, of any signature

Question

I learned in an algebra class that given any category $\mathcal{C}$ of algebras of signature $(\Omega,E)$ where $\Omega$ is the set of function symbols and $E$ is the collection of identities, colimits exists in $\mathcal{C}$. Specifically, the colimit of some diagram $\mathcal{I}$ in $\mathcal{C}$ will be the quotient of the free product of objects of $\mathcal{I}$.

The quotient is related to some co-equalizer or equalizer. However, is there instead a nice explicit description of the generators of the quotient in terms of the arrows in $\mathcal{I}$?

If such a description exists, in the special case where $\mathcal{I}$ is directed set, how does this quotient reduce to the disjoint union of objects i $\mathcal{I}$ modulo `eventual equivalence' as described in Wikipedia? https://en.wikipedia.org/wiki/Direct_limit.

If such a concrete description exists, it would be marvelous and once and for all generalize many constructions, though perhaps it might be too good to be true.

Answer 1

You've sort of given a description of generators of the colimit already: you get a generating set by taking (the images of) the elements of objects of $\mathcal{I}$.

In fact, the colimit is precisely the algebra presented by:

• Generators are the elements of objects of $\mathcal{I}$ (with the objects considered disjoint)
• For every relation between object of $\mathcal{I}$, you get the corresponding relation on the colimit
• For every map $i:I \to I'$ in $\mathcal{I}$, you get a relation $x \equiv i(x)$ for every $x \in I$

The above is pretty much exactly the characterization of a colimit in terms of coproducts and coequalizers.

Answer 2

Direct limits have lots of very nice properties, but their presentations by generators and relations are not typically very transparent. We can take as a generating system the disjoint union of a set of generators of the algebra structures whose direct limit we are looking at; and as a set of relations the union of systems of relations defining each of those objects in terms of the given generators, together with further relations saying that for each arrow$$f: A_i \to A_j$$in our directed system, each generator $x$ coming from $A_i$ is a certain expression in the generators coming from $A_j$, namely, any expression which expresses$$f(x) \in A_j$$in terms of the chosen generators of $A_j$. But the result is typically an ugly mess.

For an example that is not too bad, suppose we take the additive group of the rational numbers, regarded as the direct limit of the subgroups $(1/n)\mathbb{Z}$ over the index set of positive integers $n$, ordered by divisibility. If we take for each $(1/n)\mathbb{Z}$ the generator $x_n$ representing $1/n$, and the empty set of defining relations, then our presentation of $\mathbb{Q}$ is by the infinite set of generators $x_n$, and the relations$$x_n = mx_{mn}.$$

You've sort of given a description of generators of the colimit already: you get a generating set by taking (the images of) the elements of objects of $\mathcal{I}$.

In fact, the colimit is precisely the algebra presented by:

• Generators are the elements of objects of $\mathcal{I}$ (with the objects considered disjoint).
• For every relation between object of $\mathcal{I}$, you get the corresponding relation on the colimit.
• For every map $i:I \to I'$ in $\mathcal{I}$, you get a relation $x \equiv i(x)$ for every $x \in I$.

The above is pretty much exactly the characterization of a colimit in terms of coproducts and coequalizers.

Your description can be looked at as the special case of the one I gave, gotten by taking as generators and relations for each object "all its elements" and "all the relations that they satisfy". Your description is a canonical one, while mine requires an arbitrary choice of presentation of each of the given objects by generators.

It takes the universal property of the colimit, and expresses it as saying that the object is universal for having certain elements satisfying certain relations, which is expressed by a presentation in terms of generators and relations. So it is a natural description, despite the fact that the presentation it gives is typically messier than the sorts that one usually looks at in studying algebras described by generators and relations.