nLab limits of presheaves are computed objectwise

Idea

A basic fact of category theory says that the limit or colimit of a diagram in a category of presheaves is the presheaf whose value at any object is the limit or colimit, respectively, in the category of sets, of the values of the presheaves in the diagram, at that object.

Statement

Proposition

(limits of presheaves are computed objectwise)

Let $\mathcal{C}$ be a category and write $[\mathcal{C}^{op}, Set]$ for its category of presheaves. Let moreover $\mathcal{D}$ be a small category and consider any functor

$$F \;\colon\; \mathcal{D} \longrightarrow [\mathcal{C}^{op}, Set] \,,$$

hence a $\mathcal{D}$-shaped diagram in the category of presheaves.

Then

1. The limit of $F$ exists, and is the presheaf which over any object $c \in \mathcal{C}$ is given by the limit in Set of the values of the presheaves at $c$:

   $$\left( \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{\lim} F(d) \right)(c) \;\simeq\; \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{\lim} F(d)(c)$$

2. The colimit of $F$ exists, and is the presheaf which over any object $c \in \mathcal{C}$ is given by the colimit in Set of the values of the presheaves at $c$:

   $$\left( \underset{\underset{d \in \mathcal{D}}{\longrightarrow}}{\lim} F(d) \right)(c) \;\simeq\; \underset{\underset{d \in \mathcal{D}}{\longrightarrow}}{\lim} F(d)(c)$$

Proof

This is elementary, but we spell it out in detail.

We discuss the case of limits, the other case is formally dual.

Observe that there is a canonical equivalence

$$[\mathcal{D}, [\mathcal{C}^{op}, \Set]] \simeq [\mathcal{D} \times \mathcal{C}^{op}, Set]$$

where $\mathcal{D} \times \mathcal{C}^{op}$ is the product category.

This makes manifest that a functor $F \;\colon\; \mathcal{D} \to [\mathcal{C}^{op}, Set]$ is equivalently a diagram of the form

$$\array{ && \vdots && && \vdots \\ && \big\downarrow && && \big\downarrow \\ \cdots &\longrightarrow& F(d_1)(c_2) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_2) &\longrightarrow& \cdots \\ && \big\downarrow && && \big\downarrow \\ \cdots &\longrightarrow& F(d_1)(c_1) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_1) &\longrightarrow& \cdots \\ && \big\downarrow && && \big\downarrow \\ && \vdots && && \vdots }$$

Then observe that taking the limit of each “horizontal row” in such a diagram indead does yield a presheaf on $\mathcal{C}$, in that the construction extends from objects to morphisms, and uniquely so: This is because for any morphism $c_1 \overset{g}{\to} c_2$ in $\mathcal{C}$, a cone over $F(-)(c_2)$ induces a cone over $F(-)(c_1)$, by vertical composition with $F(-)(g)$

$$\array{ && \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ & {}^{ }\swarrow && \searrow \\ F(d_1)(c_2) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_2) \\ {}^{\mathllap{F(d_1)(g)}}\big\downarrow && && \big\downarrow^{\mathrlap{F(d_2)(g)}} \\ F(d_1)(c_1) && \overset{\phantom{AA}}{\longrightarrow} && F(d_2)(c_1) }$$

From this, the universal property of limits of sets implies that there is a unique morphism between the pointwise limits which constitutes a presheaf over $\mathcal{C}$

$$\array{ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ \big\downarrow^{\mathrlap{ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(g) }} \\ \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_1) }$$

and that is the tip of a cone over the diagram $F(-)$ in presheaves.

Hence it remains to see that this cone of presheaves is indeed universal.

Now if $I$ is any other cone over $F$ in the category of presheaves, then by the universal property of the pointswise limits, there is for each $c \in \mathcal{C}$ a unique morphism of cones in sets

$$I(c) \longrightarrow \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c) \,.$$

Hence there is at most one morphisms of cones of presheaves, namely if these components make all their naturality squares commute.

$$\array{ I(c_2) &\longrightarrow& \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_2) \\ \big\downarrow && \big\downarrow \\ I(c_1) &\longrightarrow& \underset{\underset{d \in \mathcal{D}}{\longleftarrow}}{lim} F(d)(c_1) } \,.$$

But since everything else commutes, the two ways of going around this diagram constitute two morphisms from a cone over $F(-)(c_1)$ to the limit cone over $F(-)(c_1)$, and hence they must be equal, by the universal property of limits.

Related statements

• limits and colimits by example

• limits preserve limits

• hom-functor preserves limits

• adjoints preserve (co-)limits