# Can we define factorials in sufficiently nice categories?

In sufficiently nice categories, we can define an exponential object $B^A$ for all objects $A$ and $B$. Now I tend to think of $B^A$ as an internalization of $\mathrm{Hom}(A,B)$ to the category. This begs the question: given an object $A$ living in a sufficiently nice category, is there a way to define a new object $A!$ that somehow "internalizes" the set of all isomorphisms $A \rightarrow A$?

Yes if the ambient category $\mathcal{C}$ has pullbacks (and, of course, is cartesian closed). I will denote $B^A = \underline{\mathrm{Hom}}(A,B)$ and construct a subobject $\underline{\mathrm{Isom}}(A,B)$ as follows: Consider the morphisms $$\underline{\mathrm{Hom}}(A,B) \times \underline{\mathrm{Hom}}(B,A) \to \underline{\mathrm{Hom}}(A,A) \times \underline{\mathrm{Hom}}(B,B) \longleftarrow \star$$ given by $(f,g) \mapsto (gf,fg)$ and $(id_A,id_B) \leftarrow \star$ (if the notation is not clear, ask). Let $\underline{\mathrm{Isom}}(A,B)$ be their pullback. The composition $\underline{\mathrm{Isom}}(A,B) \to \underline{\mathrm{Hom}}(A,B) \times \underline{\mathrm{Hom}}(B,A) \to \underline{\mathrm{Hom}}(A,B)$ is a monomorphism, which exhibits the subobject of isomorphisms from $A$ to $B$.
For $A=B$ I wouldn't use $A!$ as a notation, but rather $\underline{\mathrm{Aut}}(A)$.
A typical example is $\mathcal{C}=\mathsf{Sh}(X)$, the category of sheaves on a space $X$. If $A,B$ are sheaves on $X$, the isomorphism sheaf $\underline{\mathrm{Isom}}(A,B)$ is the subsheaf of the usual homomorphism sheaf $\underline{\mathrm{Hom}}(A,B)$ whose sections on $U \to X$ are the isomorphisms $A|_U \to B|_U$.
• I like your answer but I have to object that to my intuition the factorial is something different, maybe more naive: $n!=$ "what you get when you multiply $2\cdot 3\dots (n-1)\cdot n$". Your procedure obviously gives rise to the categorification of the automorphism group of a finite set, but I think we've lost the "combinatorial" side of the story. Multiplication is of course the decategorification of a monoidal product, so I would have taken a different path, albeit extremely more naive. – Fosco Jan 9 '14 at 22:30
• For example (absolutely without any desire to superimpose my opinion to yours) I think that a natural definition for $[n]^{\times }!$ in the category of simplicial sets would be the cartesian product of all representables $[n]\times [n-1]\times\dots \times [2]\times [1]\times [0]$; or even more simply in the category $\Delta$ of simplices, $[n]^\oplus!$ would be... – Fosco Jan 9 '14 at 22:33
• @tetrapharmakon, actually I called it factorial; Martin advocated calling it $\underline{\mathrm{Aut}}(A)$. Anyhow, the intent of the question was not to capture the combinatorial aspects of factorial, so much as about internalizing the group of isomorphisms of an object. – goblin Jan 10 '14 at 4:47