What does it mean for a category to be generated under (some) colimits?

Question

This is going to be a long post, so let me state my question first and then explain what I am interested in by way of examples.

Question. Is there any literature studying notions of generation under colimits and their relationships?

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In algebra, we often speak of a(n algebraic) structure $A$ being generated by a subset $A'$ if $A$ itself is the smallest subset of $A$ containing $A'$ and also closed under the given operations. Mimicking this, say:

Definition. A category $\mathcal{C}$ is 0-naïvely generated under $\Phi$-colimits by a subcategory $\mathcal{C}' \subseteq \mathcal{C}$ if it is $\Phi$-cocomplete (i.e. has all $\Phi$-colimits) and $\mathcal{C}$ itself is the smallest replete full subcategory of $\mathcal{C}$ containing $\mathcal{C}'$ and closed in $\mathcal{C}$ under $\Phi$-colimits.

Example. $\textbf{Ab}$ is 0-naïvely generated under small colimits by any subcategory containing any abelian group that contains $\mathbb{Z}$ as a direct summand.

Example. $\textbf{Vect}$ is 0-naïvely generated under small coproducts by any subcategory containing the 1-dimensional vector space.

Notice that the above definition depends only on the class of objects of $\mathcal{C}'$. We could improve it as follows:

Definition. A category $\mathcal{C}$ is 1-naïvely generated under $\Phi$-colimits by a subcategory $\mathcal{C}' \subseteq \mathcal{C}$ if $\mathcal{C}$ is $\Phi$-cocomplete and $[\mathbf{2}, \mathcal{C}]$ is 0-naïvely generated under $\Phi$-colimits by the (full) subcategory containing just the arrows in $\mathcal{C}'$.

Remark. Of course, 1-naïve generation implies 0-naïve generation: consider the diagonal embedding $\mathcal{C} \to [\mathbf{2}, \mathcal{C}]$.

Example. $\textbf{Ab}$ is 1-naïvely generated under small filtered colimits by the full subcategory of finitely generated abelian groups.

Non-example. $\textbf{Vect}$ is not 1-naïvely generated under small coproducts by the full subcategory containing the 1-dimensional vector space. (If it were, every matrix would be diagonalisable.)

Alternatively, instead of trying to fix up the naïve definition by adding conditions, we could take a more abstract approach.

Definition. A category $\mathcal{C}$ is densely generated under $\Phi$-colimits by a functor $i : \mathcal{C}' \to \mathcal{C}$ if it is $\Phi$-cocomplete and, for every $\Phi$-cocomplete category $\mathcal{D}$ and all $\Phi$-cocontinuous functors (i.e. functor that preserves $\Phi$-colimits) $F_0, F_1 : \mathcal{C} \to \mathcal{D}$, the map $$\textrm{Nat} (F_0, F_1) \to \textrm{Nat} (F_0 i, F_1 i)$$ induced by horizontal precomposition with $i$ is a bijection.

Assuming $\mathcal{C}$ is $\Phi$-cocomplete, if $i : \mathcal{C}' \to \mathcal{C}$ is absolutely dense, then $\mathcal{C}$ is densely generated under $\Phi$-colimits by $i$. On the other hand, assuming $\mathcal{C}$ is locally small with $\Phi$-cocontinuous representables $\mathcal{C} \to \textbf{Set}^\textrm{op}$ and $\textbf{Set}^\textrm{op}$ is $\Phi$-cocomplete, if $\mathcal{C}$ is densely generated under $\Phi$-colimits by $i : \mathcal{C}' \to \mathcal{C}$, then $i$ is dense. This justifies the terminology "densely".

Modulo size issues, it turns out we do not need to quantify over all $\Phi$-cocomplete categories in the above definition.

Proposition 1. Suppose $\mathcal{C}$ is locally small, $\textbf{Set}^\textrm{op}$ is $\Phi$-cocomplete, and every representable functor to $\textbf{Set}^\textrm{op}$ is $\Phi$-cocontinuous. Then the following are equivalent:

• For every locally small category $\mathcal{D}$ and all $\Phi$-cocontinuous functors $F_0, F_1 : \mathcal{C} \to \mathcal{D}$, the map $$\textrm{Nat} (F_0, F_1) \to \textrm{Nat} (F_0 i, F_1 i)$$ induced by horizontal precomposition with $i$ is a bijection.

• For all $\Phi$-cocontinuous functors $F_0, F_1 : \mathcal{C} \to \textbf{Set}^\textrm{op}$, the induced map $\textrm{Nat} (F_0, F_1) \to \textrm{Nat} (F_0 i, F_1 i)$ is a bijection.

If I'm not mistaken, we also have this:

Proposition 2. If $\mathcal{C}$ is 1-naïvely generated under $\Phi$-colimits by a full subcategory $\mathcal{C}' \subseteq \mathcal{C}$, then $\mathcal{C}$ is densely generated under $\Phi$-colimits by the inclusion $\mathcal{C}' \hookrightarrow \mathcal{C}$.

Of course, dense generation is closely related to free generation:

Definition. A category $\mathcal{C}$ is freely generated under $\Phi$-colimits by a functor $i : \mathcal{C}' \to \mathcal{C}$ if, for every $\Phi$-cocomplete category $\mathcal{D}$, the functor from the category of $\Phi$-cocontinuous functors $\mathcal{C} \to \mathcal{D}$ to the category of all functors $\mathcal{C}' \to \mathcal{D}$ induced by precomposition with $i$ is fully faithful and essentially surjective on objects.

It is clear that free generation implies dense generation. Furthermore:

Proposition 3. Let $i : \mathcal{C}' \to \mathcal{C}$ and $y : \mathcal{C}' \to \hat{\mathcal{C}}{}'$ be functors. Suppose $\hat{\mathcal{C}}{}'$ is freely generated under $\Phi$-colimits by $y$, and let $\ell : \hat{\mathcal{C}}{}' \to \mathcal{C}$ be the $\Phi$-cocontinuous functor extending $i$ along $y$. Then the following are equivalent:

• $\mathcal{C}$ is densely generated under $\Phi$-colimits by $i$.

• $\ell$ is corepresentably fully faithful (co-ff), i.e. for every $\Phi$-cocomplete category $\mathcal{D}$ and all $\Phi$-cocontinuous functors $F_0, F_1 : \mathcal{C} \to \mathcal{D}$, the map $$\textrm{Nat} (F_0, F_1) \to \textrm{Nat} (F_0 \ell, F_1 \ell)$$ induced by horizontal precomposition with $\ell$ is a bijection.

When we can construct $\hat{\mathcal{C}}{}'$ explicitly, it is usually clear that $\hat{\mathcal{C}}{}'$ is 0-naïvely generated under $\Phi$-colimits by the image of $y : \mathcal{C}' \to \hat{\mathcal{C}}{}'$. In this case, it follows that dense generation implies 0-naïve generation. What about 1-naïve generation?

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For completeness (hah) I should explain the "$\Phi$-colimit" terminology. This is supposed to be a generic nonce for colimit notions such as "small colimit", "finite colimit", "small filtered colimit", "binary coproduct", "pushout", "tensor", "splitting of an idempotent", and so on. But if I had to give a formal definition, I would start by saying that $\Phi$ is supposed to be a collection of pairs $(\mathcal{J}, W)$ where $\mathcal{J}$ is a category and $W$ is a presheaf on $\mathcal{J}$. Then a $\Phi$-colimit in a category $\mathcal{C}$ is a $W$-weighted colimit of a diagram $\mathcal{J} \to \mathcal{C}$ where $(\mathcal{J}, W)$ is in $\Phi$.

Answer 1

It seems that dense generation does not imply 1-naïve generation in general. For a counterexample it is enough to consider the class $\Phi$ for small (or just finite) coproducts.

Indeed, given any non trivial category $\mathcal C$, consider the inclusion $J\colon \mathcal C\hookrightarrow Fam(\mathcal C)$ of $\mathcal C$ into its free cocompletion under small coproducts. Recall that an object of $Fam(\mathcal C)$ is a family $(C_i)_{i\in I}$ of objects from $\mathcal C$, and a map $f=(f_i,h)\colon (C_i)_{i\in I}\to (D_j)_{j\in J}$ is determined by a function $h\colon I\to J$ together with morphisms $f_i\colon C_i\to D_{fi}$ in $\mathcal C$.

Then $J$ is dense (pretty much by definition), but $Fam(\mathcal C)$ is not 1-naïvely generated under coproducts by $\mathcal C$. This is because the closure under coproducts of $[\mathbf 2,\mathcal C]$ in $[\mathbf 2, Fam(\mathcal C)]$ has as objects those maps $f=(f_i,h)\colon(C_i)_{i\in I}\to (D_j)_{j\in J}$ for which $h$ is a bijection of sets. Thus, for instance, morphisms of the form $(C_1,C_2)\to (D)$ are not coproducts of objects from $[\mathbf 2,\mathcal C]$.