While a colimit is of a typical object in category theory hence so much in the closely related areas such as categorical logic and type theory, its construction can be a nuisance especially within a strict category setting where elementhood matters in the objects.

We describe a concrete method of “generating an element of colimit” in terms of so called *solution set* as a part of necessary and sufficient conditions of *adjoint functor theorem*.

While the idea is not original and the contents are parallel to the CWM^{[1]}MacLane, S. (1971). Categories for the Working Mathematician. New York: Springer-Verlag., we believe that the exposition is helpful and unique in the applicable interpretation.

## 1. The solution set condition and its interpretations

The *solution set condition*, originally formalized as a necessary condition that a functor (under some condition) becomes right adjoint, is an interesting assertion by itself simply because it gives an explicit construction of a colimit.

More precisely, *solution set condition* denotes the existence of “weakly universal objects” in some sense that can be expressed (or unified) in terms of functor involved with the following (general) *adjoint functor theorem* and its corollaries:

- Adjoint functor theorem;
- Condition for the existence of initial object;
- Representable functor theorem.

Although we are not giving any proofs and unnecessary only for the exposition, we denote by a locally small, complete small category. We also fix a functor where the selection of a category is the key to characterize what the solution set is for. We persist in these notations throughout the part 1.

With these in mind, the (unified) *solution set condition* is concisely stated as:

Here **wInit** denotes a (small) set of *weakly initial objects*, whose element suffices the condition of initial object except the uniqueness condition. `(*S)`

can be stated explicitly as:

While `(*S)`

is exactly the *solution set condition* for adjoint functor theorem, the condition will be interpreted accordingly in a context.

### 1.1. Condition for the existence of initial object

By setting , dropping and letting the constant functor, we obtain a necessary *condition for the existence of initial object*, namely:

Notice that canonically preserves every (small) limits because any universal cone collapses to the zero object in .

### 1.2. Representable functor theorem

This is analogous to the *condition for the existence of initial object*.

For , a representation of is identified with the universal arrow , or equivalently the *unit of adjoint* , where is identified with the faithful subcategory and the composition of followed by , furthermore a choice functor.

Therefore is representable if and only if and preserves small limit.

## 2. Subobjects and spanning maps

For an arbitrary category , we can consider “subobjects” of an object , whose elements are monos together with the order relation defined by if and only if there exists some and a mono such that , or to say concisely factors through . By defining , we call the set of equivalence classes *subobjects of a*. We denote *subobjects of a* by .

**Definition** **2.1**. For a given functor , a map is said to **span a** when there is no (non-isomorphic) mono such that factors through .

**Lemma** **2.2**. Fix a category and its object .

Suppose the pullbacks for any set of subobjects exists, namely . If a functor preserves any of such pullbacks, then every map factors through a map that spans .

**Proof** **of lemma 2.2**. For , let be a set of subobjects such that with for each i. Since by assumption there exist the pullback which we denote as , we see it suffices , where is the induced map on pullback diagrams of .

apparently spans by the construction of ; precisely, if there is a mono such that factors through for some (i.e. ), then must be in and since is the pullback, must coincide with , analogously with and with .

□

## 3. A general method of constructing a colimit as the result of a left adjoint functor from **Set**

It is an immediate consequence of *adjoint functor theorem* that we have a **general method** to construct a colimit of certain type in a particular class of categories.

The applicable class of categories represented by should suffice the following properties `(*AFT)`

:

- is locally small and complete small;
- It is given a continuous functor to ;
- suffices solution set condition
`(*S)`

.

Not exclusive yet particularly important instance of such classes is the *algebraic system* of fixed type, which forms a category of algebra of given type . suffices `(*AFT)`

along with the functor known as forgetful functor .

If we are given such functor with these properties, *adjoint functor theorem* states that the continuous functor admits a left adjoint, which preserves a colimit of , hence the image of the left adjoint is the constructed colimit in .

Moreover, this construction procedure can be done in an algorithmic manner. Above lemma shows that, the solution set of can be identified with the union set of maps that span from . This can be seen as follows:

To a given map , choosing a map through which factors is equivalent to inducing a (part of) pullback map in the diagrams of the form .

This last description roughly speaks of algorithmic nature of a (co)limit preserving condition of procedure (i.e. functor), namely providing with a “basis” in the target category against an arbitrary object of source category.

Here we mean by “basis” is a set of objects that admit irreducibleness of some sort (c.f. might be preferable to express as “atomicity” in a context).

Here is an example.

### 3.1. Coproduct in **Grp**

In , the forgetful functor admits the free group construction as the left adjoint, where a solution set of is composed of every monomorphisms of the form that has no factoring monos through a subgroup . Hence plays a part of freely generating (as a group) set for .

The free construction preserves colimit (direct sum) of , hence the coproduct is justified to have the form in , meaning that an element of is obtained by (finite applications of) group *operations *and *identities *on the base set in terms of universal algebra.

Footnotes

↑1 | MacLane, S. (1971). Categories for the Working Mathematician. New York: Springer-Verlag. |
---|