A left almost semigroup (LA-semigroup) is a groupoid \((L, m)\) consisting of a nonempty set \(L\) and a binary operation \(m\) on \(L\) satisfying \(m(l_1, m(l_2,l_3))=m(m(l_3,l_2),l_1)\) for all \(l_1,l_2,l_3\in L\). Let P(\(L\)) denote the set of all continuous mappings of the unit interval \(I=[0,1]\) into a topological LA-semigroup \(L\). An L-mapping is a continuous mapping \(f\colon L\to L'\) between topological LA-semigroups such that \(f(m(l_1,l_2))=m'(f(l_1),f(l_2))\) for all \(l_1,l_2\in L\). Two L-mappings \(h, h'\colon L\to L'\) between topological LA-semigroups are called L-homotopic if there is an L-mapping \(H\colon L\to P(L')\) with \(H(l)(0)=h(l)\) and \(H(l)(1)=h'(l)\) for all \(l\in L\). An L-mapping \(f\colon L\to L'\) between topological LA-semigroups is called an L-fibration if for every topological LA-semigroup \(L''\), an L-mapping \(g\colon L''\to L\) and an L-homotopy \(H\colon L''\to P(L')\) with \(H_0=f\circ g\), there is an L-homotopy \(G\colon L''\to P(L)\) such that \(G_0=g\) and \(f[G(l'')(x)]=H(l'')(x)\) for every \(l''\in L'', x\in I\). The authors prove that certain restrictions of L-fibrations and the composition of two L-fibrations are L-fibrations. For an L-mapping \(f\colon L\to L'\) of topological LA-semigroups, an L-lifting function is defined as a certain function from \(\{(l, \omega)\in L\times P(L')\colon f(l)=\omega(0)\}\) into \(P(L)\) and proved that an L-mapping has an L-lifting function if and only if it is an L-fibration. The concepts of L-regular fibration, L-absolute retract, L-absolute neighborhood retract and L-homotopy extension property are introduced and some of their properties are derived. For example, if \(L\) is an L-absolute retract, then P(L) is also an L-absolute retract.