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

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?

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?

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 $$\Phi$$.

• This blog post explains various notions of generators and some of their relations : qchu.wordpress.com/2015/05/17/generators Commented May 23, 2023 at 7:43
• Small notational question: does $\mathbf 2$ mean the category $\bullet \to \bullet$? (You seem to use it like an arrow, but to me that category is denoted $[1]$ and $\mathbf 2$ maybe means the discrete category $\mathbf 1 \amalg \mathbf 1$.) Commented May 23, 2023 at 9:41
• Yes, I mean $\{ \bullet \to \bullet \}$. For me, the discrete category would be $2$ (like the set). Commented May 23, 2023 at 9:49
• The notion of 1-naive generation looks pretty weird to me, and as observed in Giacomo's answer it isn't even implied by dense or free generation, so it seems questionable to me to regard it as a notion of "generation". Do you have some application in mind for it? Commented Jun 11, 2023 at 5:47
• I don’t have an application. But it’s a bit of a strawman anyway (not for nothing I called it “naïve”). Commented Jun 11, 2023 at 5:56

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 $$[\mathbf 2,\mathcal C]$$.