The relationship between toposes and set theories was studied comprehensively in

> Steve Awodey, Carsten Butz, Alex Simpson, Thomas Streicher: [Relating first-order set theories, toposes and categories of classes](https://www.sciencedirect.com/science/article/pii/S0168007213000730).
> Annals of Pure and Applied Logic, Volume 165, Issue 2, February 2014, Pages 428-502

Regularity is discussed under the name "well-foundedness". You can find a lot of details in the paper, I am just going to quickly review the setup.

We work in an elementary topos $\mathcal{E}$.

The first step is to cook up a notion of $\in$-membership.
Define a *membership graph* to be a triple $G = (|G|, A_G, r_G)$ where $|G|$ an $A_G$ are objects and $r_G : |G| \to A_G + P|G|$ a morphism. We think of $|G|$ as a set of vertices with each vertex $x \in |G|$ being either an atom $a$ (in case $r(x) = \mathrm{inl}(a)$ for $a : A_G$) or a branching vertex with adjacency set $d \subseteq |G|$ (in case $r(x) = \mathrm{inr}(d)$ for $d : P|G|$). We may define a bisimilarity relation $\sim_{G,H}$ between two membership graphs expressing the fact that, up to reordering and repetition, $G$ and $H$ represent the same $\in$-membership structure. In other words, $\sim_{G,H}$ expresses extensional equality of sets represented by $G$ and $H$. In particular, $\sim_{G,G}$ is an equivalence relation which quotients the membership graph to give a membership relation that is extensional.

Next, we define a new topos $\mathcal{E}_\mathrm{nwf}$ whose objects are triples $(D, m, G)$ where $G$ is a membership graph in $\mathcal{E}$, $D$ an object of $\mathcal{E}$, and $m : D \to |G|$ a mono in $\mathcal{E}$. A suitable notion of morphism is devised that takes into account the bisimilarity relations $\sim_{G,H}$. We then have:

**Theorem 11.7:** *$\mathcal{E}_\mathrm{nwf}$ is equivalent to $\mathcal{E}$.*

We may understand the theorem as saying that we enriched the topos $\mathcal{E}$ with membership relations to get $\mathcal{E}_\mathrm{nwf}$, and that the enrichment only change the topos up to equivalence (so not in any essential way from the point of view of topos theory).

So far we allow both atoms and non-well-founded membership relation. The next step is to define what it means for a membership graph $G = (|G|, A_G, r_G)$ to be well-founded. Here there are no surprises, as we can use the internal language of the topos to state when $X : P|G|$ satisfies the property "$X$ contains all the atoms $A_G$ and is hereditarily closed under the membership relation $r_G$" (see the formula after Corollary 11.2).

We define a third topos $\mathcal{E}_\mathrm{wf}$ as the full subcategory of $\mathcal{E}_\mathrm{nwf}$ of those objects whose membership graphs are well-founded.

**Proposition 11.3:** *The equivalence between $\mathcal{E}$ and $\mathcal{E}_\mathrm{nwf}$ cuts down to an equivalence between $\mathcal{E}$ and $\mathcal{E}_\mathrm{wf}$.*

The moral of the story is that, firstly, we may define a notion of extensional membership relation on objects in a topos, and secondly, that restricting to the well-founded part of the topos does not change the topos in a way that is relevant to topos theory. In a sense the answer to the question "when does a topos satisfy the axiom of regularity" is "it does not matter".

Disclaimer: please do not take the above summary as a satisfactory description. The paper contains many more details and explanations, and it should be consulted for thorough understanding of the topic.