# Concretely calculating the Day convolution of $\text{Cat}$-valued presheaves on a monoidal category

### Summary

Let $$(\mathcal{C}, \otimes, e)$$ be a monoidal category, and consider the category $$[\mathcal{C}^\text{op}, \text{Cat}]$$ of $$\text{Cat}$$-valued presheaves and natural transformations. I want to:

1. Find an specialized definition of Day convolution for $$[\mathcal{C}^\text{op}, \text{Cat}]$$; and
2. Find an explicit/concrete description of it.

### Finding a specialized definition

I'm aware of the standard definition of Day convolution in terms of $$\mathcal{V}$$-enriched categories. I have also seen the coend formula when $$\mathcal{V} = \text{Set}$$. However, I do not know any enriched category theory and so cannot at the moment make use of the standard definition.

I also know that $$\text{Cat}$$-enriched categories are the same as (strict) $$2$$-categories, but I haven't found any definitions of Day convolution for $$2$$-categories (which hopefully would be easier to learn than enriched category theory).

In conclusion, I'm looking for a pre-existing specialized definition to $$\mathcal{V} = \text{Cat}$$ (and thus $$\mathcal{C}$$ is given the standard/trivial enrichment as $$\text{Cat}$$ is cocomplete) or a $$2$$-categorical definition.

### Finding an explicit/concrete description

I've successfully applied the definition of Day convolution in the case of $$\text{Set}$$-valued presheaves. I expanded the coend formula to what it means in terms of universal properties. I also wrote it down as an explicit coequalizer.

In the end I had a description of the resulting set in terms of an equivalence relation generated by the two actions of the morphism on the profunctor I took the coend over.

The above is the level of concreteness I'm looking for.

### What I'd like

For (1) I'd appreciate one of

• a link if a definition exists; or
• a quick working out of the general definition in the special case of $$\text{Cat}$$ if it's trivial for you; or
• telling me I should just invest my time in learning enough enriched category theory to understand the standard definition.

For (2) if someone has a definition, an explicit description of the resulting category (objects and morphisms only is fine).

As a follow up, I spent some time learning enriched category theory, enough to take enriched coends. Thus, I could determine what Day convolution was as a $$\text{Set}$$-valued map from $$[\mathcal{C}^\text{op}, \text{Cat}] \times [\mathcal{C}^\text{op}, \text{Cat}] \to [\mathcal{C}^\text{op}, \text{Cat}]$$, which was enough for now.