Definition of simplicially enriched category

Question

In Philip Hirschhorn's Model Category and Their Localizations, Def. 9.1.2, pg. 159,

He defines a simplicially enriched category as

A category $M$ together with

1. Every two objects $X,Y$ corresponds a simplicial set $Map(X,Y)$.
   2. (Composition)
   3. (identity)
   4. Every two objects $X,Y$ of $M$ $$Map(X,Y)_0 \cong M(X,Y)$$
   5. (standard associativitiy and identity compatibility diagrams).

What confuses me is that the definition on nlab I read for category enriched over a monoidal category $K$ we declare the hom objects to be objects in $K$.

Hirschhorn's definition supposes we begin a with a category.

May someone elaborate on the difference?

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It seems to me that these are the same - Hirschhorn describes the underlying category. Am I right?

Answer 1

An enriched category $K$ over $V$ in the nLab's sense always gives rise to an ordinary category $K_0$ by defining $K_0(x,y)=V(I,K(x,y))$, where $I$ is the unit object in $V$. So it is redundant to start with a category structure, but it's more intuitive.

Answer 2

A (plain) category has a set of morphisms $M(X,Y)$; axiom 4 requests that these morphisms are the 0-simplices of the simplicial set $Map(X,Y)_\bullet : \Delta^o \to \bf Set$

Answer 3

Expanding a little bit on Kevin Carlson's answer: basically you have two ways to regard an enriched category (over a monoidal category $K$).

The first one is the n-lab way: an enriched category is made of a set of objects, a family of objects in $K$ (the hom-objects) and a family of morphisms in $K$ satisfying certain conditions.

The other one is what I call the more intuitive way: an enriched category is an ordinary category with an enrichment, that is a category whose hom-sets and composition operations are the images, along a forgetful functor, of objects and morphisms in $K$.

The two definition agree.

Every monoidal category has a natural forgetful functor to $\mathbf{Set}$ (namely the functor $K[I,-]\colon K \to \mathbf{Set}$), and the images of the $K$-hom-objects and $K$-compositions via this functor form the hom-sets and composition operations of an ordinary category, called the underlying category of the enriched category.

On the other hand an enrichment of a category consists of data ($K$-hom-objects and compositions) that form a n-lab enriched category.

Personally I tend to prefer the second definition which is closer to the intuition that an enriched category should be a category with an enrichment, i.e. an additional structure. This helped me a lot when I had to understand things like $K$-enriched natural transformation and enriched yoneda. Anyway this is a matter of tastes, so probably other people would prefer the other definition that is conceptually simpler.

I hope this help.