# Tensor product by UMP. Is this 'embeddable' in the theory of categories?

In this question $\mathbf{Ab}$ denotes the category that has abelian groups as objects and the grouphomomorphisms between them as arrows.

Definition:

For ordered pair $\left(A,B\right)$ with $A,B\in\mathbf{Ab}$ a tensorproduct is a pair $\left(T,\phi\right)$ with $T\in\mathbf{Ab}$ and $\phi:A\times B\rightarrow T$ biadditive, such that for every $C\in\mathbf{Ab}$ and biadditive $f:A\times B\rightarrow C$ there is a unique $\hat{f}\in\mathbf{Ab}\left(T,C\right)$ with $f=\hat{f}\circ\phi$.

In many UMP situations uniqueness can be expressed by saying that $\phi$ is epic, but here I miss the categorical context needed for it. Map $\phi$ can - as far as I can see - not be interpreted as some arrow in some category. My question(s):

Is my look too restricted here? Is it still possible to find a context in which $\phi$ is an arrow (and in this case an epimorphism)? If not then isn't this a shortcoming of the theory of categories?

• Could you please state which book is it? – magma Jan 21 '14 at 19:58
• @magma. It is my own concept of tensorproduct of abelian groups and does not come from a book. In an effort to emphasize the essence of it I left out the usual notation $\otimes$. It leaves open how you construct $(T,\phi)$. The existence of it is enough. – drhab Jan 21 '14 at 21:25
• oh, sorry @drhab, I misread "look" for "book" in your question – magma Jan 22 '14 at 1:04
• @magma. Ego te absolvo. – drhab Jan 22 '14 at 9:25
• @nik Universal Mapping Property. See math.stackexchange.com/q/225018/75923 – drhab Jan 22 '14 at 12:54

You probably don't mean an arbitrary $\phi$, but rather the universal example $\otimes : A \times B \to A \otimes B$. It is an epimorphism in the following sense: If $A \otimes B \rightrightarrows C$ are two homomorphisms which equalize $\otimes$, then they are equal.
You can define a multicategory whose $n$-fold morphisms are $n$-multilinear maps $A_1 \times \dotsc \times A_n \to B$. Then $\otimes$ is an epimorphism in this multicategory.
• I am familiar with the usual construction of $A\otimes B$ and $\otimes:A\times B\rightarrow A\otimes B$. There is an isomorphism $\hat{\otimes}:T\rightarrow A\otimes B$ with $\otimes=\hat{\otimes}\circ\phi$ so up to isomorphisms $\left(T,\phi\right)$ and $\left(A\otimes B,\otimes\right)$ are the same. The notion of multicategories is new to me and will hopefully fill the gap that I am speaking of in my question. – drhab Jan 4 '14 at 16:24