Computing Limits in Concrete Categories

The definition of a limit is a universal property, not a formula. In $\mathbf{Set}$ and in categories of structured sets a single two-step recipe supplies the formula: take the limit of underlying sets, then put the unique compatible structure on it. This is why limits in $\mathbf{Pos}$, $\mathbf{Mon}$, $\mathbf{Grp}$, and $\mathbf{Top}$ all look like limits in $\mathbf{Set}$, and why colimits in those categories do not.

Threads: the limit formula in Set

Fix a diagram in $\mathbf{Set}$: a shape with nodes $i,j,k,\dots$ and edges $e:i\to j$, instantiated by sets $A(i)$ and functions $A(e):A(i)\to A(j)$. If a limit $L$ exists, the representable description forces its elements:

and a cone with one-point vertex is a choice of one element per node, compatible with every edge.¹

A choice function picks one element from each set; a thread picks them compatibly, so that traveling along any edge carries the selected element to the selected element. When the shape has no edges every choice function is a thread, and the set of threads is the full product ${\prod }_{i}A(i)$.

The thread set is a subset of the product ${\prod }_{i}A(i)$ cut out by one equation per edge — the concrete face of the products-and-equalizers generation theorem. Instantiating the shape recovers every formula seen so far:

The lifting recipe

Now let $\mathcal{C}$ be a category of structured sets with a forgetful functor $U:\mathcal{C}\to \mathbf{Set}$ — posets and monotone maps, monoids and homomorphisms, spaces and continuous maps. Given a diagram in $\mathcal{C}$, apply $U$ to get a diagram in $\mathbf{Set}$, and compute its thread set $L$. The remaining work is always the same three steps.³

Downstairs in $\mathbf{Set}$ only one function can serve as the mediator, so the only open question is whether that function respects the structure. Each category contributes one idea to the furnish step.

Pos: pointwise order

For a diagram of posets and monotone maps, order the threads pointwise:

Antisymmetry is inherited from the components. Each evaluation $\alpha (i)$ is monotone by construction, and the mediator $\mu$ is monotone because each $\xi (i)$ is: $x\le y$ gives $\xi (i)(x)\le \xi (i)(y)$ for every $i$, which is exactly $\mu (x)\le \mu (y)$ in the pointwise order.⁴

Mon: componentwise operations

Threads over a diagram of monoids multiply componentwise, $(a\star b)(i)=a(i)b(i)$, with unit the thread of units $1(i)=1_i$. The product of two threads is again a thread because each $A(e)$ is a homomorphism:

Associativity and the unit laws hold pointwise, the evaluations are homomorphisms by definition of $\star$, and the mediator is a homomorphism because every $\xi (i)$ is.⁵ The same componentwise furnishing works for $\mathbf{Grp}$, $\mathbf{Ab}$, $\mathbf{Ring}$, ${\mathbf{Vect}}_k$, and any similar algebraic category: in ${\mathbf{Vect}}_k$, for instance, $(x_I{)}_I+(y_I{)}_I=(x_I+y_I{)}_I$.

Top: the projection-generated topology

For a diagram of spaces, topologize the thread set by declaring, for each node $i$ and each open $U\subseteq A(i)$, the preimage $\alpha (i{)}^{-1}(U)$ to be open, and taking these as a subbase. This is the smallest topology making every evaluation continuous. Smallness is what makes the mediator continuous: for a subbasic open set,

which is open since $\xi (i)$ is continuous — and subbasic preimages are all one needs to check.⁶ The two conditions pull in opposite directions: continuity of the projections needs the topology large, continuity of mediators needs it small, and the generated topology is the unique equilibrium. For a discrete shape this is precisely the product topology, whose familiar smallest topology with continuous projections definition is thus a special case of a limit computation.

A worked pullback in Top

Take the cospan in $\mathbf{Top}$, writing $J=[0,2\pi )$ for the half-open interval:

• Underlying set. Threads for the corner shape are pairs, so $P=\{(\theta ,x)\mid s(\theta )=t(x)\}$, i.e. pairs with $x=\theta +2\pi n$ for some integer $n$.
• Topology. The subbase is generated by the two projections into $J$ and $\mathbb{R}$, which is the subspace topology from the product $J\times \mathbb{R}$.
• Result. $P$ is a disjoint union of copies of $J$ indexed by $n\in \mathbb{Z}$ — the pullback splits the real line into the sheets it wraps over the circle, one component per winding.

Creation of limits

In each case the $\mathbf{Set}$-limit admitted exactly one structure making the projections structure-preserving, and the resulting cone was a limit cone upstairs. That is the definition of creating limits; the forgetful functors of $\mathbf{Pos}$, $\mathbf{Mon}$, $\mathbf{Grp}$, $\mathbf{Ring}$, ${\mathbf{Vect}}_k$ all create limits, and since $\mathbf{Set}$ is complete, all these categories are complete with limit-preserving forgetful functors.⁷

$\mathbf{Top}$ sits just outside the pattern: the underlying set of a limit of spaces is the limit of underlying sets, but the topology is not unique — any topology between the generated one and the discrete one keeps the projections continuous, and only the generated one is universal. The forgetful functor $\mathbf{Top}\to \mathbf{Set}$ preserves limits without creating them. The distinction is invisible when computing (the recipe still says use the generated topology) but it changes the formal status of the lift:

On the colimit side the recipe collapses. The underlying set of a coproduct of groups is not the disjoint union of underlying sets, and the initial object of $\mathbf{Grp}$ (one element) does not sit over the initial object of $\mathbf{Set}$ (empty). Colimits of algebras exist, but they must be built with free constructions and quotients, not lifted from below — except in the directed case, where eventual agreement computes them elementwise after all. The forgetful functor preserves limits but not colimits, and the reason is adjunctions, developed in interactions between functors and limits and completed in right adjoints preserve limits.