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I have some ideas about how to define morphism of morphisms in category theory, but I don't know anything about higher category theory. The idea is quite simple, and can be iterated easily. I'll only focus on Sets, but it is clear that the following concept could easily be categorifiable.

Let's give us some algebraic structure (magma, monoid, group, vector space, $\ldots$), and let $f : A \rightarrow B$ be a morphism of such structures. Notice that the graph of $f$, $\Gamma(f) = \{ (x, f(x)) \}$ inherits automatically of the same structure as $A$ and $B$, simply by defining the binary law as $(x,f(x)) * (y, f(y)) = (x*_A x', f(x) *_B f(y))$.

Let $f : A \rightarrow B$ and $g : C \rightarrow D$ two morphisms of the same type of structure. Call 2-morphism $\alpha : f \rightarrow g$ any morphism $\alpha : \Gamma(f) \rightarrow \Gamma(g)$, that is, any morphism of their graph. Clearly, $\Gamma(\alpha)$ also inherits of the same structure as $A$, $B$, $C$, and $D$, allowing us to iterate the process.

Is it known ? If not, does this sound a good idea to consider higher morphisms ?

sure
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1 Answers1

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As it turns out, $\Gamma(f)$ is simply isomorphic to $A$. Let $\varphi: A \to \Gamma(f)$ be defined by $\varphi(a) = (a, f(a))$. Then $$\varphi(a*b) = (a*b, f(a*b)) = (a,f(a)) * (b,f(b)) = \varphi(a) * \varphi(b).$$ And $\varphi$ has an obvious inverse, namely the projection on the first coordinate. Finally $A \cong \Gamma(f)$.

If you want to consider "morphisms of morphisms", there are various possibilities. One of them is the arrow category $\mathsf{Ar}(\mathsf{C})$ for example, where objects of $\mathsf{Ar}(\mathsf{C})$ are morphisms $f : A \to B$ in $\mathsf{C}$, and morphisms of $\mathsf{Ar}(\mathsf{C})$ are commutative diagrams. For example a morphism from $(f : A \to B)$ to $(g : C \to D)$ looks like: $$\require{AMScd} \begin{CD} A @>{f}>> B \\ @VVV @VVV \\ C @>{g}>> D \end{CD}$$

This isn't really related to higher category theory, though.

Najib Idrissi
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  • Ah indeed, quite simply even. Yet, isn't the concept interesting to have a geometric interpretation of the algebraic structures ? For example, in the case of $k$-vector space, the graph of a linear map $f : A \rightarrow B$ is nothing else than a subspace of $dim A$ of $AxB$. That is, for real vector space, its a ``plane'' of dim $A$ in $A \times B$. – sure Nov 06 '14 at 10:14
  • In particular this means that the number of non-trivial embedding of $A$ into $A\times B$ is nothing else than the number of (non trivial) morphisms from $A$ to $B$. – sure Nov 06 '14 at 10:21
  • One can even be tempted to define a morphism $f : A \rightarrow B$ as any embedding of $A$ inside $A \times B$. In the case of vector spaces, the set of morphisms defined as above from $A$ to $B$ gives you a Grassmanian. (Yet you might lose composition and some properties if you don't forbid infinite slope morphisms) – sure Nov 06 '14 at 10:31
  • @sure I don't understand what you're trying to say, but consider that the data of a morphism $A \to A \times B$ is equivalent to the data of two morphisms $A \to A$ and $A \to B$; for this to be considered some kind of graph you probably want $A \to A$ to be the identity, and then you're just left with a morphism $A \to B$. – Najib Idrissi Nov 06 '14 at 12:56