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limits of presheaves are computed objectwise

Contents

Context

Category theory

Limits and colimits

Contents

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 ๐’ž\mathcal{C} be a category and write [๐’ž op,Set][\mathcal{C}^{op}, Set] for its category of presheaves. Let moreover ๐’Ÿ\mathcal{D} be a small category and consider any functor

F:๐’ŸโŸถ[๐’ž op,Set], F \;\colon\; \mathcal{D} \longrightarrow [\mathcal{C}^{op}, Set] \,,

hence a ๐’Ÿ\mathcal{D}-shaped diagram in the category of presheaves.

Then

  1. The limit of FF exists, and is the presheaf which over any object cโˆˆ๐’žc \in \mathcal{C} is given by the limit in Set of the values of the presheaves at cc:

    (limโŸตdโˆˆ๐’ŸF(d))(c)โ‰ƒlimโŸตdโˆˆ๐’ŸF(d)(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 FF exists, and is the presheaf which over any object cโˆˆ๐’žc \in \mathcal{C} is given by the colimit in Set of the values of the presheaves at cc:

    (limโŸถdโˆˆ๐’ŸF(d))(c)โ‰ƒlimโŸถdโˆˆ๐’ŸF(d)(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

[๐’Ÿ,[๐’ž op,Set]]โ‰ƒ[๐’Ÿร—๐’ž op,Set] [\mathcal{D}, [\mathcal{C}^{op}, \Set]] \simeq [\mathcal{D} \times \mathcal{C}^{op}, Set]

where ๐’Ÿร—๐’ž op\mathcal{D} \times \mathcal{C}^{op} is the product category.

This makes manifest that a functor F:๐’Ÿโ†’[๐’ž op,Set]F \;\colon\; \mathcal{D} \to [\mathcal{C}^{op}, Set] is equivalently a diagram of the form

โ‹ฎ โ‹ฎ โ†“ โ†“ โ‹ฏ โŸถ F(d 1)(c 2) โŸถAA F(d 2)(c 2) โŸถ โ‹ฏ โ†“ โ†“ โ‹ฏ โŸถ F(d 1)(c 1) โŸถAA F(d 2)(c 1) โŸถ โ‹ฏ โ†“ โ†“ โ‹ฎ โ‹ฎ \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 โ€œhorizontal rowโ€ in such a diagram indead does yield a presheaf on ๐’ž\mathcal{C}, in that the construction extends from objects to morphisms, and uniquely so: This is because for any morphism c 1โ†’gc 2c_1 \overset{g}{\to} c_2 in ๐’ž\mathcal{C}, a cone over F(โˆ’)(c 2)F(-)(c_2) induces a cone over F(โˆ’)(c 1)F(-)(c_1), by vertical composition with F(โˆ’)(g)F(-)(g)

limโŸตdโˆˆ๐’ŸF(d)(c 2) โ†™ โ†˜ F(d 1)(c 2) โŸถAA F(d 2)(c 2) F(d 1)(g)โ†“ โ†“ F(d 2)(g) F(d 1)(c 1) โŸถAA F(d 2)(c 1) \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 ๐’ž\mathcal{C}

limโŸตdโˆˆ๐’ŸF(d)(c 2) โ†“ limโŸตdโˆˆ๐’ŸF(d)(g) limโŸตdโˆˆ๐’ŸF(d)(c 1) \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(โˆ’)F(-) in presheaves.

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

Now if II is any other cone over FF in the category of presheaves, then by the universal property of the pointswise limits, there is for each cโˆˆ๐’žc \in \mathcal{C} a unique morphism of cones in sets

I(c)โŸถlimโŸตdโˆˆ๐’ŸF(d)(c). 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.

I(c 2) โŸถ limโŸตdโˆˆ๐’ŸF(d)(c 2) โ†“ โ†“ I(c 1) โŸถ limโŸตdโˆˆ๐’ŸF(d)(c 1). \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(โˆ’)(c 1)F(-)(c_1) to the limit cone over F(โˆ’)(c 1)F(-)(c_1), and hence they must be equal, by the universal property of limits.

Last revised on September 26, 2018 at 11:23:48. See the history of this page for a list of all contributions to it.