[[!redirects homotopy precategories]] ## Contents ## * table of contents {:toc} ## Idea ## The oidification of an [[A3-space]] ## Definition ## A homotopy precategory $A$ consists of the following. * A type $A_0$, whose elements are called objects. Typically $A$ is coerced to $A_0$ in order to write $x:A$ for $x:A_0$. * 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 $f=1_b \circ f$ and $f=f\circ 1_a$. * For each $a,b,c,d:A$, $$f:hom_A(a,b),\ g:hom_A(b,c),\ h:hom_A(c,d)$$ we have $h\circ (g\circ f)=(h\circ g)\circ f$. ## See also ## * [[quiver]] * [[H-spaceoid]] * [[A3-space]] * [[concrete homotopy precategory]] * [[precategory]] * Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)