nLab lifted limit

Definition

Definition

Let $J\colon I \to C$ be a diagram and $F\colon C \to D$ be a functor. Let $(d, \eta)$ with $\eta \colon \text{const}_d^I \to F \circ J$ be a limit for $F \circ J$. $F$ is said to lift this limit if there exists a limiting cone $(c, \alpha)$ with $\alpha \colon \text{const}_c^I \to J$ such that $(F c, F \cdot \alpha)$ is isomorphic to $(d, \eta)$ in the category of cones over $F \circ J$. $F$ is said to lift limits for $J$ if it lifts every limiting cone of $F \circ J$.

Alternatively, this says that, if $F \circ J$ has a limit, then $J$ also has a limit and that limit is preserved by $F$. This implies $(F c, F \cdot \alpha)$ is a limiting cone, thus a terminal object in the cone category, and thus (uniquely) isomorphic to $(d, \eta)$.

Of course, all the above applies dually to colimits - $F$ lifts colimits for $J$ if and only if $F^{op} \colon C^{op} \to D^{op}$ lifts limits for the corresponding diagram $J^{op} \colon I^{op} \to C^{op}$.

Properties

Remark

If $F$ lifts a limit $(d, \eta)$ for $F \circ J$ to $(c, \alpha)$, then it necessarily lifts any limit for $F \circ J$.

Proof

Any other limit $(d', \eta')$ for $F \circ J$ is (uniquely) isomorphic to $(d, \eta)$, and thus isomorphic to $(F c, F \cdot \alpha)$. Hence, it is lifted by $F$.

Hence, lifting of limits for $J$ by $F$ either holds vacuously, or holds for all limits of $F \circ J$.

Proposition

Let $F_i \colon C \to D_i$ be a family of functors, and $J \colon I \to C$ a diagram. Then their product $F \colon C \to \prod_i D_i$ lifts limits for $J$ if and only if the components $F_i$ collectively lift limits for $J$, meaning a family of limit cones $(L_i, \lambda_{i, j})$ for $F_i \circ J$ can be simultaneously lifted to a limit cone for $J$.

Proof

We exploit limits in product categories are computed componentwise. So, limits for $F \circ J$ correspond to families of limits for $F_i \circ J$. By unfolding definitions, $F$ lifts limits for $J$ precisely if the $F_i$ collectively lift limits for $J$.

In the following, $J \colon I \to C$ is a diagram and $F \colon C \to D, G \colon D \to E$ are functors.

Proposition

Suppose $F$ lifts limits for $J$ and $G$ lifts limits for $F \circ J$. Then $G \circ F$ lifts limits for $J$.

Proof

Take a limit cone over $(G \circ F) \circ J$. Since $G$ lifts limits for $F \circ J$, this can be lifted to a limit cone for $F \circ J$ that $G$ preserves. And then, since $F$ lifts limits for $J$, this can be lifted to a limit cone for $J$ that $F$ preserves, which implies $G \circ F$ also preserves it, as requires.

Proposition

Suppose $G \circ F$ lifts limits for $J$ and $F$ preserves limits of $J$. Then $G$ lifts limits for $F \circ J$.

Proof

Take a limit cone $(z, \alpha)$ for $(G \circ F) \circ J$. Then since $G \circ F$ lifts this, we obtain a limit cone $(x, \mu)$ for $J$ that $G \circ F$ preserves. This implies $(F(x), F \cdot \mu)$ is a limit cone for $F \circ J$ which $G$ preserves, as required.

Proposition

Suppose $G \circ F$ lifts limits for $J$ and $G$ preserves and reflects limits of $F \circ J$. Then $F$ lifts limits for $J$.

Proof

Take a limit cone $(y, \mu)$ for $F \circ J$. Since $G$ preserves this, $(G(y), G \cdot \mu)$ is a limit for $(G \circ F) \circ J$. Then, since $G \circ F$ lifts this, there is a limit cone $(x, \eta)$ for $J$ preserved by $F \circ G$. Finally, since $G$ additionally reflects limits for $F \circ J$, we conclude that $F$ preserves $(x, \eta)$ too, as required.

The propositions above show that:

• If $F$ preserves limits of $J$ and lifts limits for $J$, then $G$ lifts limits for $F \circ J$ if and only if $G \circ F$ lifts limits for $J$.
• If $G$ preserves and reflects limits of $F \circ J$, and lifts limits for $F \circ J$, then $F$ lifts limits of $J$ if and only if $G \circ F$ does.

Strictness

Definition

$F$ is said to lift limits for $J$ strictly if it lifts limits for $J$ and moreover, in the notation of , $(F c, F \cdot \alpha) = (d, \eta)$.

This is often how the definition is phrased in textbooks, but it is not invariant under equivalence of categories. However, in practice many functors that lift limits do so strictly.

Definition

$F$ is said to lift limits for $J$ uniquely if it strictly lifts limits for $J$ and moreover, in the notation of , there is a unique limiting cone $(c, \alpha)$ with $(Fc, F \cdot \alpha) = (d, \eta)$.

Again, this is too strict to be invariant under equivalence of categories. If we relax strictness, then this actually holds automatically - any other limiting cone $(c', \alpha')$ with $(F c', F \cdot \alpha') \cong (d, \eta)$ is in particular a limit for $J$, and so (uniquely) isomorphic to $(c, \alpha)$.

Relation to preservation and reflection

Remark

If $F$ lifts limits for $J$, and there exists at least one limiting cone $(d, \eta)$ for $F \circ J$, then $F$ preserves limits of $J$.

Proof

This is because, in the notation of , the lift $(c, \alpha)$ is preserved by $F$, which by this remark implies all limits for $J$ are preserved by $F$.

Thus, lifting of limits either holds vacuously, or holds together with preservation of limits.

In general, reflection of limits neither implies nor is implied by lifting:

• If $F$ reflects limits for $J$, then any cone in $C$ whose image under $F$ is limiting must have already been limiting - but since this requires us to start with a cone in $C$, this cannot be used to lift a limit for $F \circ J$.

• Conversely, if $F$ lifts limits for $J$, and $(c, \alpha)$ is a cone in $C$ such that $(F c, F \cdot \alpha)$ is limiting, then there exists a limiting cone $(c', \alpha')$ for $J$ in $C$ preserved by $F$. This gives a canonical morphism $c \to c'$ induced by the universal property, but in general this may not be an isomorphism. So we cannot conclude that $(c, \alpha)$ is limiting.

Examples

• The forgetful functor $Top \to Set$ lifts limits and colimits of all small diagrams (and hence preserves them, since $Set$ is a bicomplete category). However, it does not reflect them - in general there are many possible lifts for a diagram, even strictly, by choosing coarser/finer topologies as appropriate, and not all of these can be limits/colimits.

• Any functor that creates limits necessarily lifts them. In particular, monadic functors lift limits, often strictly.

Terminological remarks

Lifting limits is closely related to creating them. The relationships between these notions were the subject of a post by Aleks Kissinger at the categories mailing list, here, but there is some dispute about its correctness.

Related concepts

• preserved limit

• reflected limit

• created limit

• topological concrete category

• lifted limit

References

See Definition 13.17 in Adamek, Herrlich?, Strecker?: Abstract and Concrete Categories. Remark 13.38 provides a useful diagram of relations between reflected/created/lifted limits.