Dear all,

Two of my joint publications with Jens Hemelaer have been published in the past week.

The first, *An Essential, Hyperconnected, Local Geometric Morphism that is not Locally Connected* is a counterexample to an open problem about geometric morphisms. Recall that the basic data of a geometric morphism consists of an adjoint pair of functors , the inverse image and direct image functors respectively, where is required to preserve finite limits, but a geometric morphism can have many extra properties, including the existence of further adjoints for the constituent functors.

There are a handful of surprising coincidences amongst properties of geometric morphisms. A geometric morphism is called *locally connected* if its inverse image preserves exponential objects and the same is true for all slices of that geometric morphism (recall that for any object of , the category is a topos, and it turns out that for as above, we get a well-defined geometric morphism whose inverse image functor sends to ). A locally connected morphism automatically has a left adjoint to . This means that a locally connected morphism is a special case of an *essential* geometric morphism.

Recall that a geometric morphism is called *connected* if its inverse image functor is full and faithful, so that the codomain topos is equivalent to a full subcategory of the domain topos. In Johnstone's 2011 paper, *Remarks on Punctual Local Connectedness*, it is observed that for a connected and locally connected geometric morphism (or indeed a connected essential geometric morphism), there is a canonical natural transformation obtained by applying full faithfulness to the composite of the counit of and the unit of . There are lots of surprising results in that paper, but the main theorem is as follows. Call a geometric morphism *punctually locally connected *if it is connected and locally connected and satisfies the innocent-sounding extra condition that is epic. Then is automatically *strongly connected* ( preserves finite products), *hyperconnected* (in addition to being full and faithful, the image of is closed under subobjects) and *local* ( has an extra right adjoint , which is automatically full and faithful). For bounded geometric morphisms, Johnstone also proves that the converse is true: a geometric morphism is punctually locally connected if and only if it is locally connected, hyperconnected and local.

To make sense of this powerful result, questions arise about how interconnected these properties are more generally. For example, we could have started with being merely essential and connected; does that make a difference? If we impose hyperconnectedness and localness, does "locally connected" come for free, or is it actually independent? That specific question is what Jens and I provided a counterexample to in our paper, as an essential geometric morphism generated by a monoid homomorphism.

The second, *Solution to a problem of FitzGerald*, is more specifically about monoids.

The most obvious source of monoids is the collection of endomorphism monoids of objects in a category. Just as understanding the automorphisms of an object can be useful, the endomorphism monoid of an object is a nice invariant to have at one's disposal. Moreover, when a category has some structure, the monoids also inherit that structure: for example, endomorphism monoids in an additive (eg abelian) category acquire an additive structure making them into rings.

As such, it would be nice to know what we can deduce about endomorphism monoids based on the properties of their ambient categories. For example, Fitzgerald observed that when a monoid of endomorphisms of an object A in a category has commuting idempotents, then there are certain properties of the retracts and coretracts of A which can be deduced from that. He conjectured that in categories of algebras (categories admitting a monadic functor to Set), this becomes an equivalence: an object satisfies the retract/coretract properties if and only if its endomorphism monoid has commuting idempotents.

That's where Jens and I come in: we spotted that one such category of algebras is the category of actions of a monoid M, and that that category contains a special object A consisting of the monoid acting on itself. It is easy to show that the endomorphism monoid of A is exactly the monoid M. So in order to produce a counterexample to Fitzgerald's claim, we just needed to construct a monoid for which the equivalence fails.

More interestingly, we showed that any counterexample to the problem (in an arbitrary category) produces such a monoid, but that the converse *might* be false. Let A be an object in a category of algebras. For three of Fitzgerald's properties, if the monoid of endomorphisms of A has that property, then so must A, but the final property does not lift to A.

Finally, we invoke some general embedding results to show that there is some ring for which the conjecture fails, and we investigate some examples, showing that in some categories, Fitzgerald's conjecture does actually hold.