The objective in this paper is four-pronged.

1.

to develop the elementary theory of isotropy groups and isotropy quotients of small categories.

2.

to present a sequence of categories \(\mathbb{X}\left[ \kappa\right] \) for each small ordinal \(\kappa\) so that the isotropy quotient sequence of \(\mathbb{X}\left[ \kappa\right] \) stabilizes after exactly \(\kappa\) steps.

3.

to obtain results about isotropy of categories arising as sequential colimits of categories and as collages of certain profunctors.

4.

to establish the fact (Proposition 9.2) that the isotropy group of a presheaf topos \(\boldsymbol{Set}^{\mathbb{C}^{\mathrm{op}}}\) coincides with that of the category \(\mathbb{C}\) [J. Funk et al., Theory Appl. Categ. 26, 660–709 (2012; Zbl 1275.18009)].

The paper consists of 10 sections, and the only background knowledge required for the paper is gladly, with the exception of §9, basic category theory. The core algebra of a Grothendieck topos [P. Freyd, Theory Appl. Categ. 18, 303–320 (2007; Zbl 1139.18001); Theor. Comput. Sci. 375, No. 1–3, 193–200 (2007; Zbl 1118.18004)] could be called the isotropy monoid of the topos, and the authors believe that the techniques and concepts developed in the paper should have interesting interpretations for Freyd’s original motivation, namely parametric polymorphisms.