We said that ordinary categories were just Set-categories, but our definition of \(\mathcal{V}\) categories required the \(\mathcal{V}\) to be a preorder!
We have to generalize (categorify) \(\mathcal{V}\) categories.
Symmetric monoidal preorders can be considered as symmetric monoidal categories, despite not providing the data for identities and composition (these are not needed because there is no choice).
Example of property becoming structure: \(I \leq \mathcal{X}(x,x)\) is a property of \(\mathcal{V}\) categories as defined earlier but become part of the structure in the categorified version of the definition.
A \(\mathcal{V}\) category (a category enriched in \(\mathcal{V}\)) where \(\mathcal{V}\) is a symmetric monoidal category.
Call the category \(\mathcal{X}\). There are four constituents:
A collection of objects, \(Ob(\mathcal{X})\)
For every pair in \(Ob(\mathcal{X})\) one specifies the hom-object \(X(x,y) \in \mathcal{V}\).
For every object, specify a \(I \xrightarrow{id_x}X(x,x) \in \mathcal{V}\) called the identity element
For every pair of compatible morphisms, a \(\mathcal{X}(x,y)\otimes\mathcal{X}(y,z)\xrightarrow{;}\mathcal{X}(x,z)\) called the composition morphism.
These satisfy the usual associative and unital laws.
Recall the example with Set as a symmetric moniodal category. Apply the definition of a \(\mathcal{V}\) category and see if this agrees with the definition of an ordinary category. Is there a subtle difference?
We’ve replaced the identity morphisms with maps from the monoidal unit, but that is functionally equivalent to ‘just picking one’ given that the initial object is a singleton.
What are identity elements in Lawvere metric spaces (Cost-categories)? How do we interpret this in terms of distances?
\(0\) was the monoidal unit - the distance from an object to itself.