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Let $B$ be a full subcategory of $C$. The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is easily seen to be equivalent to the condition that for every $c \in C$ the set of all morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of the canonical diagram $(B \downarrow c) \to B \to C$. Therefore one then calls $B$ a dense subcategory of $C$. If you just require that every $c$ is the colimit of some diagram which factors over $B$, then $B$ is called colimit-dense. This is a weaker condition: $\{R\}$ is not dense in $\mathrm{Mod}(R)$, but it is colimit dense. Note that already $\{R \oplus R\}$ is dense in $\mathrm{Mod}(R)$.

You mention that the open subsets of $\mathbb{R}^n$ constitute a dense subcategory of the smooth $n$-manifolds. But actually already $\mathbb{R}^n$ (together with all its endomorphisms) suffices since this is also the local model for the open subsets of $\mathbb{R}^n$. Consequently, we get a fully faithful embedding from the category of smooth $n$-manifolds into the category of right $M$-sets, where $M$ is the monoid of all continuous maps $\mathbb{R}^n \to \mathbb{R}^n$. I don't know if this is of use at all, but I think this is an interesting point of view. It is a sort of algebraic representation from a geometric category of interest, very similar to the functor of points approach in algebraic geometry (where we take affine schemes as a dense subcategory).

You can find more about dense subcategories at the nlab and in "Abstract and Concrete Categories - The Joy of Cats" (Examples 2.11 + Exercise 12.D, online). The definition also plays a central role in the definition of locally presentable categories; see the book by Adamek and Rosicky.

Yoneda's Lemma now just asserts that $C$ is dense in $C$, what else should we expect? There are other formal similarities with the topological notion of the same name: Let $B \subseteq D \subseteq C$ be full subcategories. If $B \subseteq C$ is dense, then $D \subseteq C$ is dense. If $B \subseteq D$ and $D \subseteq C$ are dense, then $B \subseteq C$ is dense.

Let $B$ be a full subcategory of $C$. The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is easily seen to be equivalent to the condition that for every $c \in C$ the set of all morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of the canonical diagram $(B \downarrow c) \to B \to C$. Therefore one then calls $B$ a dense subcategory of $C$. If you just require that every $c$ is the colimit of some diagram which factors over $B$, then $B$ is called colimit-dense. This is a weaker condition: $\{R\}$ is not dense in $\mathrm{Mod}(R)$, but it is colimit dense. Note that already $\{R \oplus R\}$ is dense in $\mathrm{Mod}(R)$.

You mention that the open subsets of $\mathbb{R}^n$ constitute a dense subcategory of the smooth $n$-manifolds. But actually already $\mathbb{R}^n$ (together with all its endomorphisms) suffices since this is also the local model for the open subsets of $\mathbb{R}^n$. Consequently, we get a fully faithful embedding from the category of smooth $n$-manifolds into the category of right $M$-sets, where $M$ is the monoid of all continuous maps $\mathbb{R}^n \to \mathbb{R}^n$. I don't know if this is of use at all, but I think this is an interesting point of view. It is a sort of algebraic representation from a geometric category of interest, very similar to the functor of points approach in algebraic geometry (where we take affine schemes as a dense subcategory).

You can find more about dense subcategories at the nlab and in "Abstract and Concrete Categories - The Joy of Cats" (Examples 2.11 + Exercise 12.D, online). The definition also plays a central role in the definition of locally presentable categories; see the book by Adamek and Rosicky.

Yoneda's Lemma now just asserts that $C$ is dense in $C$, what else should we expect? There are other formal similarities with the topological notion of the same name: Let $B \subseteq D \subseteq C$ be full subcategories. If $B \subseteq C$ is dense, then $D \subseteq C$ is dense. [If $B \subseteq D$ and $D \subseteq C$ are dense, then $B \subseteq C$ is dense. -- This is not correct, see comment.]

Let $B$ be a full subcategory of $C$. The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is easily seen to be equivalent to the condition that for every $c \in C$ the set of all morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of the canonical diagram $(B \downarrow c) \to B \to C$. Therefore one then calls $B$ a dense subcategory of $C$. If you just require that every $c$ is the colimit of some diagram which factors over $B$, then $B$ is called colimit-dense. This is a weaker condition: $\{R\}$ is not dense in $\mathrm{Mod}(R)$, but it is colimit dense. Note that already $\{R \oplus R\}$ is dense in $\mathrm{Mod}(R)$.

You mention that the open subsets of $\mathbb{R}^n$ constitute a dense subcategory of the smooth $n$-manifolds. But actually already $\mathbb{R}^n$ (together with all its endomorphisms) suffices since this is also the local model for the open subsets of $\mathbb{R}^n$. Consequently, we get a fully faithful embedding from the category of smooth $n$-manifolds into the category of right $M$-sets, where $M$ is the monoid of all continuous maps $\mathbb{R}^n \to \mathbb{R}^n$. I don't know if this is of use at all, but I think this is an interesting point of view. It is a sort of algebraic representation from a geometric category of interest, very similar to the functor of points approach in algebraic geometry (where we take affine schemes as a dense subcategory).

You can find more about dense subcategories at the nlab and in "Abstract and Concrete Categories - The Joy of Cats" (Examples 2.11 + Exercise 12.D, online). The definition also plays a central role in the definition of locally presentable categories; see the book by Adamek and Rosicky.

Let $B$ be a full subcategory of $C$. The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is easily seen to be equivalent to the condition that for every $c \in C$ the set of all morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of the canonical diagram $(B \downarrow c) \to B \to C$. Therefore one then calls $B$ a dense subcategory of $C$. If you just require that every $c$ is the colimit of some diagram which factors over $B$, then $B$ is called colimit-dense. This is a weaker condition: $\{R\}$ is not dense in $\mathrm{Mod}(R)$, but it is colimit dense. Note that already $\{R \oplus R\}$ is dense in $\mathrm{Mod}(R)$.

You mention that the open subsets of $\mathbb{R}^n$ constitute a dense subcategory of the smooth $n$-manifolds. But actually already $\mathbb{R}^n$ (together with all its endomorphisms) suffices since this is also the local model for the open subsets of $\mathbb{R}^n$. Consequently, we get a fully faithful embedding from the category of smooth $n$-manifolds into the category of right $M$-sets, where $M$ is the monoid of all continuous maps $\mathbb{R}^n \to \mathbb{R}^n$. I don't know if this is of use at all, but I think this is an interesting point of view. It is a sort of algebraic representation from a geometric category of interest, very similar to the functor of points approach in algebraic geometry (where we take affine schemes as a dense subcategory).

You can find more about dense subcategories at the nlab and in "Abstract and Concrete Categories - The Joy of Cats" (Examples 2.11 + Exercise 12.D, online). The definition also plays a central role in the definition of locally presentable categories; see the book by Adamek and Rosicky.

Yoneda's Lemma now just asserts that $C$ is dense in $C$, what else should we expect? There are other formal similarities with the topological notion of the same name: Let $B \subseteq D \subseteq C$ be full subcategories. If $B \subseteq C$ is dense, then $D \subseteq C$ is dense. If $B \subseteq D$ and $D \subseteq C$ are dense, then $B \subseteq C$ is dense.

Let $B$ be a full subcategory of $C$. The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is easily seen to be equivalent to the condition that for every $c \in C$ the set of all morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of the canonical diagram $(B \downarrow c) \to B \to C$. Therefore one then calls $B$ a dense subcategory of $C$. If you just require that every $c$ is the colimit of some diagram which factors over $B$, then $B$ is called colimit-dense. This is a weaker condition: $\{R\}$ is not dense in $\mathrm{Mod}(R)$, but it is colimit dense. Note that already $\{R \oplus R\}$ is dense in $\mathrm{Mod}(R)$.

You mention that the open subsets of $\mathbb{R}^n$ constitute a dense subcategory of the smooth $n$-manifolds. But actually already $\mathbb{R}^n$ (together with all its endomorphisms) suffices since this is also the local model for the open subsets of $\mathbb{R}^n$. Consequently, we get a fully faithful embedding from the category of smooth $n$-manifolds into the category of right $M$-sets, where $M$ is the monoid of all continuous maps $\mathbb{R}^n \to \mathbb{R}^n$. I don't know if this is of use at all, but I think this is an interesting point of view. It is a sort of algebraic representation from a geometric category of interest, very similar to the functor of points approach in algebraic geometry (where we take affine schemes as a dense subcategory).

You can find more about dense subcategories at the nlab and in Exercise 12.D of "Abstract and Concrete Categories - The Joy of Cats" (online). The definition also plays a central role in the definition of locally presentable categories; see the book by Adamek and Rosicky.

Let $B$ be a full subcategory of $C$. The condition that $C \to \mathrm{Set}^{C^{op}} \to \mathrm{Set}^{B^{op}}$ is fully faithful is easily seen to be equivalent to the condition that for every $c \in C$ the set of all morphisms from objects in $B$ to $c$ is a colimit diagram. In other words, $c$ is the colimit of the canonical diagram $(B \downarrow c) \to B \to C$. Therefore one then calls $B$ a dense subcategory of $C$. If you just require that every $c$ is the colimit of some diagram which factors over $B$, then $B$ is called colimit-dense. This is a weaker condition: $\{R\}$ is not dense in $\mathrm{Mod}(R)$, but it is colimit dense. Note that already $\{R \oplus R\}$ is dense in $\mathrm{Mod}(R)$.

You mention that the open subsets of $\mathbb{R}^n$ constitute a dense subcategory of the smooth $n$-manifolds. But actually already $\mathbb{R}^n$ (together with all its endomorphisms) suffices since this is also the local model for the open subsets of $\mathbb{R}^n$. Consequently, we get a fully faithful embedding from the category of smooth $n$-manifolds into the category of right $M$-sets, where $M$ is the monoid of all continuous maps $\mathbb{R}^n \to \mathbb{R}^n$. I don't know if this is of use at all, but I think this is an interesting point of view. It is a sort of algebraic representation from a geometric category of interest, very similar to the functor of points approach in algebraic geometry (where we take affine schemes as a dense subcategory).

You can find more about dense subcategories at the nlab and in "Abstract and Concrete Categories - The Joy of Cats" (Examples 2.11 + Exercise 12.D, online). The definition also plays a central role in the definition of locally presentable categories; see the book by Adamek and Rosicky.