Why are simplicial categories useful?

Question

By simplicial category here I mean simplicially enriched category, i.e. all $Hom$-sets are simplicial sets and compositions are morphisms of simplicial sets.

My question is the following. Suppose I have a category, which I can simplicially enrich. Why would that be useful? What can I do with this extra-structure? What useful information about my category I can obtain using that?

I know that I can compute homotopy groups of $Hom$-sets. Then fundamental group of $Hom(X,Y)$ will classify morphisms between $X$ and $Y$ up to homotopy. I am not sure what will higher groups mean. And I am not sure if it is the most natural thing to do with simplicial category.

Thank you very much for your help!

Answer 1

Well, with simplicial sets you can do homotopy theory, and they are closely related to topological spaces (via the adjunction given by the geometric realization and the singular set). Whereas plain categories are just abstract algebraic gadgets, simplicial categories are really geometric objects. For example, as you have already observed, one can look at the homotopy groups of the hom sets.

Here is a basic (and sort of universal) example: If $n \in \mathbb{N}$, we have a simplicial category $\mathfrak{C}[n]$ with objects $\{0,\dotsc,n\}$, and the simplicial set of morphisms $i \to j$ is the nerve of the poset $\{J \subseteq \{i,\dotsc,j\} : i \in J, j \in J\}$. It is a good exercise to write down the composition of the morphisms, etc. By abstract nonsense, we then get a functor $\mathfrak{C}[-]$ which maps each simplicial set $K$ to some simplicial category $\mathfrak{C}[K]$, and this functor is left adjoint to some simplicial nerve functor. Some details can be found in the beginning of Lurie's HTT. There you also find that simplicial categories (whose morphism sets are Kan) model $(\infty, 1)$-categories.

Answer 2

This is very natural! In fact, simplicially enriched categories are one of the models for $(\infty, 1)$-categories. So while you might not get more information about the underlying ordinary category – and in fact you can't, because every category is simplicially enriched in a trivial way – simplicial enrichment means the category has some higher-dimensional structure!

The archetypical simplicially enriched category is, of course, the category of all simplicial sets, $\mathbf{sSet}$. The simplicial set of morphisms $X \to Y$ is defined to be the simplicial set $[X, Y]$ whose $n$-simplices are morphisms $\Delta^n \times X \to Y$. It is then straightforward to see that $\pi_0 [X, Y]$ is the set of simplicial homotopy classes of morphisms $X \to Y$. (Note, not the fundamental group!)

It is not hard to check that $\pi_0 : \mathbf{sSet} \to \mathbf{Set}$ preserves finite products. It follows that we can define a category $\pi_0 [\mathbf{sSet}]$ whose objects are simplicial sets and whose set of morphisms $X \to Y$ is the set $\pi_0 [X, Y]$. Unfortunately, this is not the Quillen homotopy category $\operatorname{Ho} \mathbf{sSet}$: it is merely the localisation of $\mathbf{sSet}$ at the simplicial homotopy equivalences, rather than the weak homotopy equivalences. On the other hand, if $\mathbf{Kan}$ is the full subcategory of Kan complexes, then $\pi_0 [\mathbf{Kan}]$ does turn out to be the localisation of $\mathbf{Kan}$ at the weak homotopy equivalences. You can even define weak homotopy equivalences in $\mathbf{sSet}$ this way: a morphism $f : X \to Y$ in $\mathbf{sSet}$ is a weak homotopy equivalence if and only if the induced maps $$\pi_0 [f, K] : \pi_0 [Y, K] \to \pi_0 [X, K]$$ are bijections for all Kan complexes $K$.

A good starting point for studying homotopy theory in simplicially enriched categories is [Goerss and Jardine, Simplicial homotopy theory].

Answer 3

I'd like to point out to @Sasha that what Zhen Lin explained comes from his notes, which I found really useful and well-written from the CT point of view.

Also, once you will be more familiar with the constructions in homotopical algebra and simplicial homotopy theory, I warmly invite you to read E. Riehl's notes on Categorical Homotopy Theory, the seminal paper by C. Rezk, and the extremely clear account written by J. Bergner.

My answer is nothing more than a tentative to thank all these people for their continuous effort in spreading these beautiful and deep ideas, so that even a childish ignorant like I am can hope to understand, one day, these topics.