# nLab H-category

### Context

#### Higher algebra

higher algebra

universal algebra

## Idea

The oidification of an A3-space/H-monoid?, or equivalently, a category internal to $Ho(Top)_*$.

## Definition

### Classically

A H-category or A3-spaceoid is a category internal to the classical homotopy category of topological spaces Ho(Top), or in the homotopy category $Ho(Top)_*$ of pointed topological spaces.

### Internally in an (infinity,1)-category

For the time being see category object in an (infinity,1)-category#HCategoryTypes.

### In homotopy type theory

In homotopy type theory, a H-category or A3-spaceoid is a type $A$ with

• For each $a,b:A$, a type $hom_A(a,b)$, whose elements are called arrows or morphisms.

• For each $a:A$, a morphism $1_a:hom_A(a,a)$, called the identity morphism.

• For each $a,b,c:A$, a function

$hom_A(b,c) \to hom_A(a,b) \to hom_A(a,c)$

called composition, and denoted infix by $g \mapsto f \mapsto g \circ f$, or sometimes $gf$.

• For each $a,b:A$ and $f:hom_A(a,b)$, we have the left unitor $\lambda_f:f=1_b \circ f$ and the right unitor $\rho_f:f=f\circ 1_a$.

• For each $a,b,c,d:A$,

$f:hom_A(a,b),\ g:hom_A(b,c),\ h:hom_A(c,d)$

we have the associator $\alpha_{f, g, h}:h\circ (g\circ f)=(h\circ g)\circ f$.