CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
hom-set, hom-object, internal hom, exponential object, derived hom-space
loop space object, free loop space object, derived loop space
The free loop space $\mathcal{L}X$ of a topological space $X$ (based or not) is the space of all loops in $X$. This is in contrast to the based loop space of a based space $X$ for which the loops are at the fixed base point $x_0\in X$.
The free loop space carries a canonical action (infinity-action) of the circle group. The homotopy quotient by that action $\mathcal{L}(X)/S^1$ (the “cyclic loop space”) contains what is known as the twisted loop space of $X$.
For $X$ a topological space, the free loop space $L X$ is the topological space $Map(S^1,X)$ of continuous maps in compact-open topology.
If we work in a category of based spaces, then still the topological space $Map(S^1,X)$ is in the non-based sense but has a distinguished point which is the constant map $t\mapsto x_0$ where $x_0$ is the base point of $X$.
If $X$ is a topological space, the free loop space $L X$ of $X$ is defined as the free loop space object of $X$ formed in the (∞,1)-category Top.
See at Sullivan model of free loop space.
loop space object, free loop space object,
loop space, free loop space, derived loop space
Kathryn Hess, Free loop spaces in topology and physics, pdf, slides from Meeting of Edinburgh Math. Soc. Glasgow, 14 Nov 2008
D. Ben-Zvi, D. Nadler, Loop spaces and connections, arxiv/1002.3636