Why are 1-morphisms functors

category-theoryhigher-category-theory

I'm learning Higher Category Theory but there's one thing I can't wrap my head around $n$-morphisms.

If a $0$-morphism is an object, why is a $1$-morphism a functor and not just a regular morphism?

From what I've been reading, a $1$-category is a category with $0$-morphisms and $1$-morphisms, yet other sources imply that morphisms are not the same as functors in general.

So, what is the difference between a morphism and a functor? Are they both $1$-morphisms? If so, how does that work?

My intuition tells me that morphisms could be defined as such:
$f = 1_Cf$ where $1_C$ is the identity functor in the cagetory $C$. It follows then that functors are a generalization of morphisms, so a morphism is a functor, but a functor is not a morphism. However I got no clue.

Best Answer

Take small categories $\mathcal{C}$ and $\mathcal{D}$, which are objects of $\mathbf{Cat}$, then the functors $\mathcal{C}\rightarrow\mathcal{D}$ are the morphisms of $\mathbf{Cat}$. The class $\operatorname{Fct}(\mathcal{C},\mathcal{D})$ or $\operatorname{Cat}(\mathcal{C},\mathcal{D})$ of functors $\mathcal{C}\rightarrow\mathcal{D}$ can itself be made into a category again, denoted by $\underline{\operatorname{Fct}}(\mathcal{C},\mathcal{D})$ or $\underline{\operatorname{Cat}}(\mathcal{C},\mathcal{D})$, by now considering the functors as the objects and the natural transformations between them as the morphisms. Therefore functors can be objects as well as morphisms depending on what category you consider and caution is necessary! In literature, according to my experience so far, the term "morphism" is more preferably used as it is more general, but "functor" is, when you need to be precise like here.

Combining both of these categorical structures together makes $\mathbf{Cat}$ into a $2$-category and choosing the notation of $n$-morphism avoids any confusion: Small categories are the $0$-morphisms, the functors between them are the $1$-morphisms and the natural transformations between them are the $2$-morphisms.

If you are going into the direction of $\infty$-categories, for which there are many models, the most common being certain simplicial sets, then the best way to imagine an $n$-morphism is by taking the partially ordered sets $[n]=\{0,\ldots,n\}$, which induce categories. You can now notice:

  • An object (more concretely a $0$-morphism) is equivalently a functor $[0]\rightarrow\mathcal{C}$ and therefore $\operatorname{Ob}\mathcal{C}=\operatorname{Cat}([0],\mathcal{C})$.
  • A morphism (more concretely a $1$-morphism) is equivalently a functor $[1]\rightarrow\mathcal{C}$ and therefore $\operatorname{Ar}\mathcal{C}=\operatorname{Cat}([1],\mathcal{C})$. Be cautious that the morphism is not the functor, but you have basically only rewritten your data, so considering it as such is basically identical.
  • Taking the face maps $d_0^1\colon[0]\rightarrow[1],0\mapsto 1$ and $d_1^1\colon[0]\rightarrow[1],0\mapsto 0$, you can use it to express the domain $\operatorname{dom}(f)=\mathrm{d}_1^1\circ f$ and codomain $\operatorname{cod}(f)=\mathrm{d}_1^1\circ f$ of any $1$-morphism $f$.

We can now extend this: Letting $n$-morphisms be the functors $[n]\rightarrow\mathcal{C}$ or $\operatorname{Cat}([n],\mathcal{C})$, which for example makes $2$-morphisms the commutating triangles of the category in particular, is called the nerve construction, which is a functor $N\colon\mathbf{Cat}\rightarrow\mathbf{sSet}$. It is fully faithful (Proposition 3.11.), which corresponds to the fact, that you keep the $0$- and $1$-morphisms the way they are and add informations only depending on them and their composition. You can also use general face maps $d_i^n\colon[n-1]\rightarrow[n]$ (order-preserving and leaving out $i$) as well as general degeneracy maps $s_i^n\colon[n+1]\rightarrow[n]$ (order-preserving and hitting $i$ twice) for generalisations of the domain and codomain. This makes an $n$-morphism basically connect $n+1$ $(n-1)$-morphisms.

By the way: Helpful literature about higher category theory is Higher Categories and Homotopical Algebra by Denis-Charles Cisinski (found here) or Kerodon by Jacob Lurie (found here as interactive website or here as a script).

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