13.3 Presheaves and the category of presheaves

Presheaves are contravariant functors from a category to Set, assigning sets to objects and functions to morphisms. They generalize the notion of a family of sets indexed by a category, providing a flexible framework for studying structures that vary over different contexts.

The category of presheaves on a category C is itself a rich mathematical structure called a topos. This category has objects as presheaves and morphisms as natural transformations, allowing for powerful constructions and connections to other areas of mathematics.

Presheaves

Definition and examples of presheaves

• A presheaf $F$ on a category $C$ is a contravariant functor $F: C^{op} \to Set$ that assigns a set to each object in $C$ and a function between sets to each morphism in $C$
  • For each object $X$ in $C$, $F$ assigns a set $F(X)$ (stalks)
  • For each morphism $f: X \to Y$ in $C$, $F$ assigns a function $F(f): F(Y) \to F(X)$ (restriction maps) that reverses the direction of the morphism
  • $F$ preserves identity morphisms and composition ensuring consistency and compatibility of the assigned sets and functions
• Examples of presheaves illustrate the concept:
  • For a topological space $X$, the assignment of open sets $U \subseteq X$ to the set of continuous functions $U \to \mathbb{R}$ (sheaf of continuous functions) is a presheaf on the category of open sets of $X$
  • For a group $G$, the assignment of each subgroup $H \leq G$ to the set of left cosets $G/H$ (coset space) is a presheaf on the category of subgroups of $G$

Construction of presheaf categories

• The category of presheaves on a category $C$, denoted $\hat{C}$ or $PSh(C)$, is a category whose objects are presheaves on $C$ and morphisms are natural transformations between presheaves
  • Objects: Presheaves on $C$, i.e., contravariant functors $F: C^{op} \to Set$
  • Morphisms: Natural transformations $\alpha: F \to G$ between presheaves $F$ and $G$
    • A natural transformation $\alpha: F \to G$ assigns to each object $X$ in $C$ a function $\alpha_X: F(X) \to G(X)$ such that for every morphism $f: X \to Y$ in $C$, the following diagram commutes ensuring compatibility of the assigned functions:

      F(Y) --F(f)--> F(X)
       |              |
      α_Y            α_X
       |              |
       v              v
      G(Y) --G(f)--> G(X)
      

  • Composition of morphisms is the usual composition of natural transformations preserving the structure of the presheaf category
  • The identity morphism on a presheaf $F$ is the natural transformation $1_F$ with components $(1_F)_X = 1_{F(X)}$ serving as the identity for composition

Proof of presheaf category as topos

• The category of presheaves $\hat{C}$ is a topos, a category with rich structure and properties, because it satisfies the following:
  • $\hat{C}$ has all finite limits and colimits computed pointwise for each object $X$ in $C$, enabling the construction of complex objects from simpler ones
  • $\hat{C}$ has exponentials, where for presheaves $F$ and $G$, the exponential $G^F$ is defined by $(G^F)(X) = Hom_{\hat{C}}(yX \times F, G)$ with $yX$ being the Yoneda embedding of $X$, allowing for function spaces within the presheaf category
  • $\hat{C}$ has a subobject classifier $\Omega$ defined by $\Omega(X) = \{S \mid S \text{ is a sieve on } X\}$, where a sieve on $X$ is a collection of morphisms with codomain $X$ closed under precomposition, enabling the classification of subobjects

Presheaves vs representable functors

• A representable functor is a presheaf naturally isomorphic to a functor $Hom_C(-, X)$ for some object $X$ in $C$
  • $Hom_C(-, X)$ is a contravariant functor from $C$ to $Set$ that assigns the set $Hom_C(Y, X)$ to each object $Y$ and the precomposition function $Hom_C(f, X): Hom_C(Z, X) \to Hom_C(Y, X)$ to each morphism $f: Y \to Z$
• The Yoneda lemma establishes a bijection $Hom_{\hat{C}}(Hom_C(-, X), F) \cong F(X)$, natural in both the presheaf $F$ and the object $X$, connecting presheaves and representable functors
  • This bijection implies that the Yoneda embedding $y: C \to \hat{C}$, defined by $y(X) = Hom_C(-, X)$, is fully faithful, embedding $C$ into its presheaf category
• Consequently, every presheaf is a colimit (generalized limit) of representable functors
  • Specifically, for a presheaf $F$, there is a canonical isomorphism $F \cong \int^{X \in C} F(X) \cdot Hom_C(-, X)$, expressing $F$ as a coend (generalized colimit) of representable functors weighted by the sets $F(X)$