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Stream: theory: category theory

Topic: A cute definition of enriched categories

Tim Campion (Feb 18 2022 at 13:20):

Here's one way to define what an enriched category is, which I find cute:

• Let $Mon^{aug}$ be the free monoidal category on an augmented monoid $M$, with unit $I$.
• Let $\Delta^{op}_+$ be the dual augmented simplex category -- the free monoidal category on a comonoid, and let $i : \Delta^{op} \to \Delta^{op}_+$ be the inclusion.
• Let $B : \Delta^{op} \to Mon^{aug}$ be the functor (with no particular monoidal properties) which implements the bar construction $B(I,M,I)$.
• Let $Disc : Set \to Cat$ be the inclusion of the discrete categories (a cartesian monoidal functor).

Now

• A monoidal category corresponds to a strong monoidal functor $V : Mon^{aug} \to Cat$
• A set corresponds to a strong monoidal functor $X : \Delta^{op}_+ \to Set$.

And finally,

• A $V$-enriched category with object set $X$ corresponds to a natural transformation $C: Disc X i \Rightarrow V B$ (with no particular monoidal properties).

The one thing I find less cute about this definition is the way monoidal and non-monoidal structures are mixed. Is there any way to massage this definition into something even cuter?