Preservation, Reflection, and Creation of Limits

The lifting recipe leaned on one informal phrase: the forgetful functor lets you compute the limit underneath. Made precise, it splits into three inequivalent things a functor $F:\mathcal{A}\to \mathcal{B}$ can do with limits — preserve them, reflect them, create them. $\mathbf{Top}\to \mathbf{Set}$ preserves limits without reflecting them; $\mathbf{Grp}\to \mathbf{Set}$ creates them; and creation is the strongest of the three, implying the other two in the presence of limits downstairs. Two structural theorems round out the picture: representable functors preserve all limits, and limits in a functor category are computed pointwise.

Preservation and reflection

Preservation restates as a canonical isomorphism. If $D$ has a limit and $F$ preserves it, then $F\circ D$ has a limit and the canonical comparison map is invertible:

where the comparison map is the mediating arrow induced by the cone $(F(p_I){)}_I$. Preservation demands more than an abstract isomorphism of vertices (it demands that this particular map be one), but in practice the distinction is usually suppressed.

A functor preserving all small limits is called continuous (and cocontinuous for colimits); the terminology deliberately echoes analysis, where continuous maps preserve limits of sequences.

The two properties are independent, and the standard counterexample separates them. The forgetful functor $U:\mathbf{Top}\to \mathbf{Set}$ preserves both limits and colimits, but it does not reflect them: take non-discrete spaces $X,Y$ and equip the set $U(X)\times U(Y)$ with the discrete topology $Z$. The cone $X\leftarrow Z\to Y$ maps down to a genuine product cone in $\mathbf{Set}$, but it is not a product cone in $\mathbf{Top}$, because the discrete topology is strictly larger than the product topology. The underlying sets do not distinguish the two; the topologies do.²

Colimits break the symmetry for algebra. The forgetful functor $\mathbf{Grp}\to \mathbf{Set}$ does not preserve initial objects (the trivial group has one element; the initial set has none), and ${\mathbf{Vect}}_k\to \mathbf{Set}$ does not preserve binary coproducts (the direct sum is not a disjoint union). Forgetful functors out of algebraic categories almost never preserve all colimits. The deeper reason is that they have left adjoints, and adjoints preserve limits only from their own side, completed in right adjoints preserve limits.³

Creation

The two-step recipe for limits in $\mathbf{Grp}$ (compute the limit of underlying sets, then find that exactly one group structure makes the projections homomorphisms) is a stronger phenomenon than preservation, and it has its own name.

The definition as stated (Leinster's, following most textbook usage in its strict form) asks for equality $F(A)=B$ on the nose. Equality of objects is suspicious in category theory, and the healthy version relaxes it: whenever $F\circ D$ has a limit, some cone on $D$ maps to a limit cone, and every cone that does is itself a limit cone. All examples here satisfy the strict version.⁵

Creation is the strongest of the three properties in the situations that matter:

To prove it, take the limit cone of $F\circ D$ downstairs, lift it uniquely upstairs, and observe the lift is a limit by the second clause — the lifting recipe again. Creation also implies reflection. Since $\mathbf{Set}$ is complete and the forgetful functors from $\mathbf{Grp}$, $\mathbf{Ring}$, ${\mathbf{Vect}}_k$, $\mathbf{Pos}$, $\mathbf{Mon}$ create limits, all these categories are complete and their forgetful functors preserve limits: one lemma, five completeness theorems. The uniqueness half is checked exactly as in the worked $\mathbf{Grp}$ computation: if $p_1,p_2$ must be homomorphisms, the group law on the product set is forced componentwise.

The three properties, side by side:

$\mathbf{Top}\to \mathbf{Set}$ fails creation for the same reason it fails reflection: many topologies on the thread set make the projections continuous, so the lift exists but is not unique. Only the projection-generated topology is universal.

Representables preserve limits

The hom-functor$\mathcal{A}(A,-):\mathcal{A}\to \mathbf{Set}$ turns limits in $\mathcal{A}$ into limits of sets. Two isomorphisms compose to prove it. First, for any diagram $D$ with a limit, maps into the limit are cones (the representable description):

Second, a direct computation with the thread formula: an element of $\lim \mathcal{A}(A,D)$, the limit in $\mathbf{Set}$ of the diagram of hom-sets $I\mapsto \mathcal{A}(A,D(I))$, is a family $(f_I{)}_I$ of arrows with $D(u)\circ f_I=f_J$ for every edge, which is precisely a cone on $D$ with vertex $A$.⁷ Chaining the two:

For products this is the familiar bijection $\mathcal{A}(A,X\times Y)\cong \mathcal{A}(A,X)\times \mathcal{A}(A,Y)$; for equalizers it says maps into an equalizer are maps into $X$ equalized by post-composition.

Dualizing, that is replacing $\mathcal{A}$ by ${\mathcal{A}}^{\mathrm{op}}$, the contravariant representable $\mathcal{A}(-,A)$ carries colimits in $\mathcal{A}$ to limits in $\mathbf{Set}$:

e.g. $\mathcal{A}(X+Y,A)\cong \mathcal{A}(X,A)\times \mathcal{A}(Y,A)$: a map off a sum is a pair of maps. Both statements output limits: hom-sets convert every universal construction into a limit in $\mathbf{Set}$, whether the construction was a limit or a colimit in $\mathcal{A}$.⁹ The theorem also works as a negative test: a functor that fails to preserve some limit cannot be representable, and (ahead of the adjunction chapter) cannot have a left adjoint.

Limits in functor categories

Functors $\mathsf{A}\to \mathcal{S}$ form a category $[\mathsf{A},\mathcal{S}]$ with natural transformations as arrows.¹⁰ Limits there are computed pointwise, one object of $\mathsf{A}$ at a time.

Concretely, for a diagram $D:\mathsf{I}\to [\mathsf{A},\mathcal{S}]$ of functors, define a functor $L$ by

the limit in $\mathcal{S}$ of the values at $A$; a map $f:A\to B$ in $\mathsf{A}$ induces $L(f):L(A)\to L(B)$ as the mediating arrow between the two limits, since a natural transformation of diagrams always induces a map of limits. The projections assemble into natural transformations $L\to D(I)$, and universality is checked one $A$ at a time: a cone of natural transformations evaluates at $A$ to a cone in $\mathcal{S}$, factors uniquely there, and the resulting components are natural. The product of two functors $X,Y:\mathsf{A}\to \mathcal{S}$ is the functor $A\mapsto X(A)\times Y(A)$; the equalizer of two natural transformations is computed equalizer-by-equalizer; and dually for all colimits.

Two hypotheses of the theorem cannot be dropped. The smallness conditions, $\mathsf{A}$ small and $\mathcal{S}$ locally small, keep $[\mathsf{A},\mathcal{S}]$ locally small. And completeness matters: if $\mathcal{S}$ lacks some limits, the functor category can contain exotic limits that are not computed pointwise and are not preserved by the evaluations.¹² When $\mathcal{S}=\mathbf{Set}$, the theorem says presheaf categories $[{\mathsf{A}}^{\mathrm{op}},\mathbf{Set}]$ are complete and cocomplete with everything computed pointwise, the launching point for limits and colimits of presheaves.

The module in one table

The representable description of limits is itself an adjunction: the limit operation is a right adjoint to the diagonal functor, and hom-functor continuity, pointwise limits, and limit-preserving forgetful functors are all instances of the theorem that right adjoints preserve limits.