fix: fix typo

Author: mensah-j

posts/the-yoneda-lemma.md

@@ -10,7 +10,7 @@ The Yoneda lemma, according to Vakil in his _Foundations of Algebraic Geometry_
 
 In category theory, a _presheaf_ is a mild generalization of the usual geometric notion of a presheaf. Instead of attaching data to each open set of a topological space, we attach data to each object in a category. More precisely, let $\mathscr{C}$ be a small category. A **presheaf** on $\mathscr{C}$ is a functor $\mathscr{F} \colon \mathscr{C}^{\rm op} \to \mathbf{Set}$. We denote the category of presheaves on $\mathscr{C}$, with morphisms being natural transformations of functors, by $\mathbf{PSh}(\mathscr{C})$.
 
-It is instructive to examine the prototypical case, where $\mathscr{C} = \mathbf{Open}(X)$ for some topological space $X$. Here, the objects of $\mathscr{C}$ are the open sets of $X$ and the morphisms are given by the inclusions of open sets. We can view $\mathbf{Open}(X)$ as a subcategory of $\mathbf{Top}$ by corresponding each inclusion $V \subseteq U$ to its associated embedding $\iota_{V, U} \colon V \to U$. In this setting, a presheaf $\mathscr{F}$ on $X$ attaches to each open subset $U \subseteq X$ a set of _sections_ $\mathscr{F}(U)$; furthermore, for each inclusion $\iota_{V, U}$, there is a corresponding _restriction_ map $\operatorname{res}_{U, V}$ which takes a section in $\mathscr{F}(U)$ and restricts it to obtain a section in $\mathscr{F}(V)$. Any presheaf or sheaf on a topological space can be interpreted as arising from the sheaf of continuous sections to a larger space sitting above $X$, known as the étale space, $\operatorname{Et}(\mathscr{F})$. Here, sections in $\mathscr{F}(U)$ correspond to subspaces $\operatorname{Et}(\mathscr{F})$ which lie over $U$, and $\operatorname{res}_{V, U}(s)$ is just the set-theoretic restriction, which may also be written as the pullback $\iota_{V, U}^* s$.
+It is instructive to examine the prototypical case, where $\mathscr{C} = \mathbf{Open}(X)$ for some topological space $X$. Here, the objects of $\mathscr{C}$ are the open sets of $X$ and the morphisms are given by the inclusions of open sets. We can view $\mathbf{Open}(X)$ as a subcategory of $\mathbf{Top}$ by corresponding each inclusion $V \subseteq U$ to its associated embedding $\iota_{V, U} \colon V \to U$. In this setting, a presheaf $\mathscr{F}$ on $X$ attaches to each open subset $U \subseteq X$ a set of _sections_ $\mathscr{F}(U)$; furthermore, for each inclusion $\iota_{V, U}$, there is a corresponding _restriction_ map $\operatorname{res}_{U, V}$ which takes a section in $\mathscr{F}(U)$ and restricts it to obtain a section in $\mathscr{F}(V)$. Any presheaf or sheaf on a topological space can be interpreted as arising from the sheaf of continuous sections to a larger space sitting above $X$, known as the étale space, $\operatorname{Et}(\mathscr{F})$. Here, sections in $\mathscr{F}(U)$ correspond to subspaces of $\operatorname{Et}(\mathscr{F})$ which lie over $U$, and $\operatorname{res}_{V, U}(s)$ is just the set-theoretic restriction, which may also be written as the pullback $\iota_{V, U}^* s$.
 
 In the general case, we can think of $\operatorname{Hom}_{\mathscr{C}}(V, U)$ as comprising all of the ways in which the structure of $V$ can "fit inside" the structure of $U$, and the set of sections $\mathscr{F}(U)$ can be thought as representing all the ways the data of $U$ fits inside of $\mathscr{F}$, much like in the étale space point-of-view; this is the viewpoint suggested on the [nLab article on presheaves](https://ncatlab.org/nlab/show/presheaf). Given a morphism $f \in \operatorname{Hom}(V, U)$, the analog of the restriction map is the morphism $\mathscr{F}(f) \colon \mathscr{F}(U) \to \mathscr{F}(V)$. In this article, we will call this map the _pullback_ by $f$ and denote it by $f^*$ when the presheaf $\mathscr{F}$ is understood.