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Stream: theory: category theory

Topic: Enriched ind completion

Morgan Rogers (he/him) (Aug 01 2023 at 14:27):

Hi folks. Besides rig categories, my current project has dragged me into enriched category world, and I have found myself needing to lift the ind-completion (see [[ind-object]]) to the enriched setting. Searching for this, comments on a mathoverflow question directed me to @Steve Lack and @Giacomo Tendas' 2022 paper. While this is useful for getting to grips with how different notions of filtered and flat interact, I was surprised not to find an analogue of the diagrammatic construction of the ind-completion. My question, therefore, is whether there is any obstacle to the following construction.

Let $\mathcal{C}$ be a $\mathcal{V}$-category, where $\mathcal{V}$ is sufficiently complete and cocomplete for what follows to be defined. Its enriched ind-completion could have:

• as objects, the weighted diagrams $(W: \mathbb{D}^{\mathrm{op}} \to \mathcal{V}, J: \mathbb{D} \to \mathcal{C})$ such that ($\mathbb{D}$ is a small $\mathcal{V}$-category and) $W$ is ($\omega$-)flat in the sense of Definition 2.2 of Lack and Tendas' paper.
• as hom-object $\mathrm{Hom}\left((W,J), (W',J')\right) := \lim_{W^{\mathrm{op}}} \operatorname{colim}_{W'}\mathcal{C}(J^{\mathrm{op}}-,J'-)$.

I am not quite fluent enough with the translation between weighted and conical colimits in the set case to be confident in my assessment that this recovers the usual construction of the ind-completion in the $\mathrm{Set}$ case. Nonetheless, this should produce the free cocompletion under flat weighted colimits on the condition that said colimits are closed under iteration; i.e. that a flat weighted colimit in this ind-category can be translated into a flat weighted colimit over the original category.

Nathanael Arkor (Aug 01 2023 at 14:47):

If I understand your question properly, what you are looking for is the construction of the free cocompletion of an enriched category under a class of weights, as a subcategory of the enriched presheaf category (in your case, the class of weights would be the flat weights). A standard reference is Kelly–Schmitt's Notes on enriched categories with colimits of some class (see §3).

Morgan Rogers (he/him) (Aug 01 2023 at 15:00):

Thanks Nathanael! It's good to know that the procedure for abstractly proving the existence of said cocompletion lifts to the enriched case (although I imagine Kelly did this in the opposite direction), but I'm really asking whether the "formal diagram" presentation of the ind-completion lifts to the enriched setting, since it's much easier to present Day convolution in the ordinary category case using that presentation.

Giacomo (Aug 01 2023 at 15:42):

The presentation you suggest should work. First note that the ind completion of $\mathcal C$ can be seen as the full subcategory $Flat(\mathcal C^{op},\mathcal V)$, of the presheaf $\mathcal V$-category, spanned by the flat $\mathcal V$-functors. Then take $\mathcal E$ to be the free $\mathcal V$-category on the set of $(W,J)$ as in your first bullet point, and consider the induced $F\colon\mathcal E\to Flat(\mathcal C^{op},\mathcal V)$ given by $F(W,J)=colim_WZJ$ (where $Z$ is the inclusion of $\mathcal C$ into $Flat(\mathcal C^{op},\mathcal V)$). FInally, the $\mathcal V$-category you describe is given by the (identity on objects, fully faithful) factorization of $F$. SInce $F$ is essentially surjective, the resulting $\mathcal V$-category is equivalent to the Ind-completion of $\mathcal C$.

Morgan Rogers (he/him) (Aug 01 2023 at 16:21):

Thank you! That's exactly the kind of comparison I was looking for. In case anyone is interested, the presentation of Day convolution I'm referring to is by constructing the (enriched tensor) product of the indexing categories, computing the weight via the monoidal product in $\mathcal{V}$ and computing the new diagram by taking the pointwise monoidal product in $\mathcal{C}$.