nLab preserved limit

Idea

If $J\colon I \to C$ is a diagram and $x$ is a limit of it in $C$, then we may naïvely say that this limit is preserved by a functor $F\colon C \to D$ if $F(x)$ is a limit of the composite diagram $I \overset{J}\to C \overset{F}\to D$. However, it is not enough to state this at the level of objects; we also need to impose some coherence conditions, preserving the entire universal cone. Furthermore, we can use a trick involving the Yoneda embedding to get a meaningful condition even if $J$ has no limit in $C$ at all.

Definitions

Definition

Let $F\colon C \to D$ be a functor, $J\colon I \to C$ a diagram, and $\eta \colon const^I_x \to J$ be a cone over $J$. Suppose $(x, \eta)$ is a limit for $J$. The functor $F$ preserves this limit if $(F(x), F \cdot \eta)$ is a limit of $F \circ J$.

Here, $F\cdot\eta\colon \const^I_{F(x)} \to F \circ J$ is a whiskering.

Dually, $F$ preserves a colimit of $J$ if $F^\op\colon C^\op \to D^\op$ preserves it as a limit of $J^\op\colon I^\op \to C^\op$.

For instance:

• Let $I$ be the empty category, so that a limit of the unique functor $J\colon I \to C$ is a terminal object $1$. Then $F$ preserves this terminal object if and only if $F(1)$ is a terminal object of $D$.

• Let $I$ be the discrete category $\mathbf{2}$, so that $J$ picks out two objects $a$ and $b$ of $C$ and the limit of $J$ is a product $a \times b$ of $a$ and $b$. Note that this product comes equipped with product projections $\pi\colon a \times b \to a$ and $\rho\colon a \times b \to b$. Then $F$ preserves this product if and only if $F(a \times b)$ is a product of $F(a)$ and $F(b)$ and furthermore the product projections are $F(\pi)$ and $F(\rho)$.

Definition

If $F$ preserves all limits/colimits for a specified diagram $J\colon I \to C$, we say that $F$ preserves limits of $J$.

If $F$ preserves all limits or colimits of a given type (i.e. over a given category $I$), we simply say that $F$ preserves that sort of limit (e.g. $F$ preserves products, $F$ preserves equalizers, etc.).

Definition

A functor that preserves all small limits in $C$ that exist is called a continuous functor.

Usually this term is only used when $C$ has all small limits, i.e. is a complete category.

Properties

Proposition

If there exists at least one limit $(x, \eta)$ for $J$ that is preserved by $F$, then every limit for $J$ is preserved by $F$.

Proof

Any other limit $(y, \mu)$ is uniquely isomorphic to $(x, \eta)$, and $F$ preserves this isomorphism (as all functors do). This implies $(F(y), F \cdot \mu)$ is a limit for $F \circ J$.

Thus, for a specified diagram $J\colon I \to C$, preservation of limits either holds vacuously (if $J$ has no limit in $C$), or holds for all limit cones over $J$.

Proposition

Suppose $(x, \eta)$ is a limit for $J$ and $(y, \mu)$ is a limit for $F \circ J$. Since $(F(x), F \cdot \eta)$ is a cone over $F \circ J$, there is a canonical comparision map $f \colon F(x) \to y$ with $(F \cdot \eta)_j = \mu_j \circ f$ for each $j \in I$. Then $F$ preserves $(x, \eta)$ if and only if this comparision map is an isomorphism.

Proof

$F$ preserves $(x, \eta)$ if and only if $(F(x), F \cdot \eta)$ is a limit for $F \circ J$. Since $(y, \mu)$ is already a limit for $F \circ J$, that is if and only if $(F(x), F \cdot \eta)$ is isomorphic to $(y, \mu)$ in the category of cones over $F \circ J$. And since $(y, \mu)$ is terminal in this category, that’s if and only if the comparison map, between the unique morphism $(F(x), F \cdot \eta) \to (y, \mu)$ in the cone category, is an isomorphism.

Proposition

Let $F_i \colon C \to D_i$ be a family of functors, and $J \colon I \to C$ a diagram. Then every $F_i$ preserves limits of $J$ if and only if their product $F \colon C \to \prod_i D_i$ preserves limits of $J$.

Proof

We exploit limits in product categories are computed componentwise. If every $F_i$ preserves a limit $(x, \eta)$ of $J$, then the $(F_i(x), F_i \cdot \eta)$ collectively determine a limit $(F(x), F \cdot \eta)$, so that $F$ preserves this. Conversely, if $F$ preserves the limit, then limits being computed componentwise implies each $F_i$ preserves the limit.

For the following properties, we have $J \colon I \to C$ a diagram, and $F \colon C \to D, G \colon D \to E$ functors.

Proposition

If $F$ preserves limits of $J$ and $G$ preserves limits of $F \circ J$, we have that $G \circ F$ preserves limits of $J$.

Proof

Take a limit $(x, \eta)$ for $J$. Then $(F(x), F \cdot \eta)$ is a limit for $F \circ J$ since $F$ preserves this limit, and then $((G \circ F)(x), (G \circ F) \cdot \eta)$ is a limit for $(G \circ F) \circ J$ since $G$ preserves this limit, as required.

Proposition

If $G \circ F$ preserves limits of $J$ and $G$ reflects limits of $F \circ J$, we have that $F$ preserves limits of $J$.

Proof

Take a limit $(x, \eta)$ for $J$. We want to show $(F(x), F \cdot \eta)$ is a limit for $F \circ J$. Since $G$ reflects limits of $F \circ J$, it suffices to show $((G \circ F)(x), (G \circ F) \cdot \eta)$ is a limit for $(G \circ F) \circ J$. But this follows since $G \circ F$ preserves limits of $J$.

Proposition

If $G \circ F$ preserves limits of $J$ and $F$ lifts limits for $J$, we have that $G$ preserves limits of $F \circ J$.

Proof

Take a limit $(y, \mu)$ for $F \circ J$. We want to show $(G(y), G \cdot \mu)$ is a limit for $(G \circ F) \circ J$. Since $F$ lifts limits for $J$, there is some limit cone $(x, \eta)$ over $J$ with $(F(x), F \cdot \eta) \cong (y, \mu)$ in the category of cones over $F \circ J$. Thus, $((G \circ F)(x), (G \circ F) \cdot \eta) \cong (G(y), G \cdot \mu)$ in the category of cones over $(G \circ F) \circ J$. But the former is a limit for $(G \circ F) \circ J$ since $G \circ F$ preserves limits of $J$, implying the latter is too.

The propositions above taken together imply the following:

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

Examples

• limits preserve limits

• hom-functors preserve limits

• adjoints preserve (co-)limits

• the Yoneda embedding preserves limits (see there)

Non-Examples

Let $A$ be any infinite set, and consider the constant functor $\Delta_A \colon C \to Set$ sending every object of $C$ to $A$, and every morphism to the identity.

Then this functor preserves binary products objectwise, since $\Delta_A(x \times y) = A \cong A \times A \cong \Delta_A(x) \times \Delta_A(y)$. However, it does not preserve the product of $x$ and $y$ in the sense above, since the projection maps $\pi_x \colon x \times y \to x, \pi_y \colon x \times y \to y$ get sent to identities $A \to A$, which do not form a limit cone in $Set$.

This illustrates that it is not enough to merely show that $F(\lim J)$ is a limit of $F \circ J$ at the level of objects - we also need that the projections get mapped to the corresponding projections.

We can also view this through the language of representable functors. Preserving limits “objectwise” means $F(\lim J)$ represents the presheaf $d \mapsto Cones(d, F \circ J) := Nat(\Delta_d, F \circ J)$ on $D$, so that there is a natural isomorphism $D(-, F(\lim J)) \cong Cones(-, F \circ J)$. However, for $F$ to preserve the limit we don’t just care that $F(\lim J)$ represents this functor, but how it does - in other words, we need the corresponding universal element in $Cones(F(\lim J), F \circ J)$ to be $F$ applied to the projections in $Cones(\lim J, J)$. Given the natural isomorphism, then, we can check $F$ preserves the limit by following the identity $F(\lim J) \to F(\lim J)$ and checking it maps to the correct element.

Preservation of weighted limits

Analogously, an enriched functor between enriched categories may preserve weighted limits. Are there any tricky points that we should mention?

Preservation of limits that don't exist

Sometimes we want to say that a functor $F\colon C \to D$ preserves a limit that does not actually exist in $C$. For instance, a finitely continuous functor is usually defined as one that preserves all finite limits. If $C$ is a finitely complete category, then this is fine; such a functor is called left exact. But what if $C$ does not have all finite limits?

If $C$ and $D$ are locally small, then we can use the Yoneda lemma to turn the question into one involving categories that do have the required limits (and in fact have all limits), the presheaf categories $[C^op,Set]$ and $[D^op,Set]$. (For colimits, use $[C,Set]$ and $[D,Set]$; for $V$-enriched categories, use $[C^op,V]$ and $[D^op,V]$, which will work if $V$ is complete.)

The left Kan extension of the composite $C \overset{F}\to D \overset{Yon}\hookrightarrow [D^\op,Set]$ along the Yoneda embedding $C \overset{Yon}\hookrightarrow [C^\op,Set]$ (which always exists) is a functor from $[C^op,Set]$ to $[D^op,Set]$, which may be written as $- \otimes F$ (alluding to the bimodule nature of profunctors). A diagram $J\colon I \to C$ becomes a diagram $I \overset{J}\to C \overset{Yon}\hookrightarrow [C^op,Set]$ in $[C^op,Set]$, where it has a limit. If $- \otimes F$ preserves this limit, then we say that $F$ preserves the hypothetical limit of $J$.

Since the Yoneda embedding preserves and reflects all limits, if $J$ has a limit in $C$, then this condition is equivalent to the condition that $F$ preserve it in the ordinary sense, but in general it is stronger than requiring that $F$ preserve the limit only if it exists in $C$.

Finishing the motivating example, a flat functor may be defined as one that preserves all finite limits, whether or not they exist.

Related concepts

• (co)continuous functor

• reflected limit

• created limit

• lifted limit

References

• Saunders MacLane, §V.4 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Francis Borceux, §2.4 in: Handbook of Categorical Algebra Vol. 1: Basic Category Theory [doi:10.1017/CBO9780511525858]

• Emily Riehl, §3.3 in: Category Theory in Context, Dover Publications (2017) [pdf, book website]