In 1967 *Steenrod* wrote what later became a quite celebrated paper, **[A convenient category of topological spaces][1]** *(Michigan Math. J. 14 (1967) 133–152)*. The paper conveys the work of many (among the most significant, the [PhD thesis][2] of *Ronnie Brown*) and provides a description of a [convenient category of topological spaces][3]. Among the requirements, we see ***cartesian closedness***. Since then, several papers studying *categories of spaces* like topoi, or locales, have characterised exponentiable objects, having in mind that seminal paper of Steenrod and it is impossible to recall all the papers impacted by that work. In a sense, even the [gros-toposes][4] used in synthetic [differential][5] and algebraic geometry have among their main features to be cartesian closed, connecting the call of Steenrod to the general framework of [axiomatic cohesion][6] (and its [relatives][7]). I always found this requirement very useful. Of course many constructions come more natural, or just easier, when one can juggle with an internal hom. Recently though I have been wondering what actually motivated Steenrod, whether he has something very concrete in mind, or was guided from an *aesthetic* need. > **Q.** In which proofs, theorems, or main constructions of general or algebraic topology (possibly also algebraic geometry) is it **important or very useful** that the homset actually carry a structure of topological spaces? List of applications of internal homs I am aware of. - *Exa. 1.* Higher homotopy groups can be reduced to $\pi_0$ via loop-spacing. - *Exa. 2.* Similarly cohomology, which can be phrased in terms of cohomotopy. [1]: https://projecteuclid.org/journals/michigan-mathematical-journal/volume-14/issue-2/A-convenient-category-of-topological-spaces/10.1307/mmj/1028999711.full [2]: https://ora.ox.ac.uk/objects/uuid:3af55800-4be7-462f-b91d-9769a6dac2c4 [3]: https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces [4]: https://ncatlab.org/nlab/show/big+and+little+toposes [5]: https://en.wikipedia.org/wiki/Synthetic_differential_geometry [6]: https://ncatlab.org/nlab/show/cohesive+topos [7]: https://mathoverflow.net/questions/356618/what-is-the-precise-relationship-between-pyknoticity-and-cohesiveness?rq=1