I was studying the axioms of a category, and noted that one axiom says there is an element $1\_X\in Hom(X,X)$ for any object $X$ which serves as the identity. Why is this axiom necessary? What happens if I drop this axiom?




---


Background: I can define the category of affine holomorphic symplectic varieties, by saying 


1. The objects are semisimple algebraic groups
2. The morphisms $Hom(G,G')$ are affine holomorphic symplectic varieties with Hamiltonian $G\times G'$ action
3. The composition of two morphisms $X\in Hom(G,G')$ and $Y\in Hom(G',G'')$ is given by the holomorphic symplectic quotient $X\times Y//G'$.


This becomes a nice symmetric monoidal category; the identity in $Hom(G,G)$ is $T^\*G$. 


Suppose I want to consider the category of hyperkähler manifolds instead. I can try the following 


1. The objects are semisimple compact groups
2. The morphisms $Hom(G,G')$ are hyperkähler manifolds with Hamiltonian $G\times G'$ action
3. The composition of two morphisms $X\in Hom(G,G')$ and $Y\in Hom(G',G'')$ is given by the hyperkähler quotient $X\times Y///G'$.


Now, the problem is that $T^\*G\_\mathbb{C}$ has a hyperkähler metric (constructed by [Kronheimer](http://arxiv.org/abs/math.DG/0409253)) and almost acts like an identity, but not quite: given a hyperkähler manifold $X$ with $G$ action, $T^\*G\_\mathbb{C} \times X /// G$ is equivalent to $G$ as holomorphic symplectic varieties but not equivalent as hyperkähler manifolds. 


What should I do?




---


For my purpose, I guess using the terminology semigroupoid would suffice (I just want to define the target "category" of a TQFT precisely.) But I'm curious what kind of hell will break loose if I drop this axiom, why the people who originally defined categories included this into the axiom, etc.



Why does Hom need an identity in the definition of the category?
Your structure can be described as a "category without identity", which has been given the names "semicategory" and "semigroupoid" presumably due to independent discoveries.


Some Googling suggests the term "semicategory" came first, in [a 1972 TAMS paper by Mitchell](http://www.ams.org/journals/tran/1972-167-00/S0002-9947-1972-0294441-0/S0002-9947-1972-0294441-0.pdf). The name is motivated by applying an analogy connecting groups and semigroups to categories (as categories without identities or inverses), and it seems to be popular among people who study categories.


The term "semigroupoid" seems to have appeared first in Tilson, *Categories as Algebra: an essential ingredient in the theory of monoids* in *Journal of Pure and Applied Algebra* **48** (1987) 83-198. The name is motivated by applying an analogy connecting groups and groupoids to semigroups (as semigroups with multiple objects), and it seems to be popular among people who study semigroups.


I think the analogy is a bit weak on the semicategory side, since categories don't straightforwardly generalize groups. I'm not in charge, though.


John Baez points out in his [TWF week 296](http://math.ucr.edu/home/baez/week296.html) that there is a canonical way to make a category out of a semicategory, by formally adding, for each object, an identity element to the set of morphisms from that object to itself (and preserving all other morphism sets and composition laws). Any previously existing identities become idempotents. He notes that the categories formed this way are distinguished among all categories by the property that all invertible morphisms are identities. In particular, this process of formally adding identities is reversible in a canonical way, and no hell will break loose.


