The synovium is a mesenchymal tissue composed mainly of fibroblasts with a lining and sublining that surrounds the joints. In rheumatoid arthritis (RA), the synovial tissue undergoes marked hyperplasia, becomes inflamed and invasive and destroys the joint 1 , 2 . Recently, we and others found that a subset of fibroblasts located in the sublining undergoes major expansion in RA and is linked to disease activity 3 , 4 , 5 . However, the molecular mechanism by which these fibroblasts differentiate and expand in RA remains unknown. Here, we identified a critical role for NOTCH3 signaling in the differentiation of perivascular and sublining CD90( THY1 )+ fibroblasts. Using single cell RNA-sequencing and synovial tissue organoids, we found that NOTCH3 signaling drives both transcriptional and spatial gradients in fibroblasts emanating from vascular endothelial cells outward. In active RA, NOTCH3 and NOTCH target genes are markedly upregulated in synovial fibroblasts. Importantly, genetic deletion of Notch3 or monoclonal antibody-blockade of NOTCH3 signaling attenuates inflammation and prevents joint damage in inflammatory arthritis. Our results indicate that synovial fibroblasts exhibit positional identity regulated by endothelium-derived Notch signaling and that this stromal crosstalk pathway underlies inflammation and pathology in inflammatory arthritis.

Diffeological and differential spaces are generalisations of smooth structures on manifolds. We show that the "intersection" of these two categories is isomorphic to Frölicher spaces, another generalisation of smooth structures. We then give examples of such spaces, as well as examples of diffeological and differential spaces that do not fall into this category.We apply the theory of diffeological spaces to differential forms on a geometric quotient of a compact Lie group. We show that the subcomplex of basic forms is isomorphic to the complex of diffeological forms on the geometric quotient. We apply this to symplectic quotients coming from a regular value of the momentum map, and show that diffeological forms on this quotient are isomorphic as a complex to Sjamaar differential forms. We also compare diffeological forms to those on orbifolds, and show that they are isomorphic complexes as well.We apply the theory of differential spaces to subcartesian spaces equipped with families of vector fields. We use this theory to show that smooth stratified spaces form a full subcategory of subcartesian spaces equipped with families of vector fields. We give families of vector fields that induce the orbit-type stratifications induced by a Lie group action, as well as the orbittype stratifications induced by a Hamiltonian group action.

Abstract. If a Lie group acts on a manifold freely and properly, pulling back by the quotient map gives an isomorphism between the differential forms on the quotient manifold and the basic differential forms upstairs. We show that this result remains true for actions that are not necessarily free nor proper, as long as the identity component acts properly, where on the quotient space we take differential forms in the diffeological sense.

This paper exhibits equivalences of 2-stacks between certain models of S 1gerbes and differential 3-cocycles. We focus primarily on the model of Dixmier-Douady bundles, and provide an equivalence between the 2-stack of Dixmier-Douady bundles and the 2-stack of differential 3-cocycles of height 1, where the 'height' is related to the presence of connective structure. Differential 3-cocycles of height 2 (resp. height 3) are shown to be equivalent to S 1 -bundle gerbes with connection (resp. with connection and curving). These equivalences extend to the equivariant setting of S 1 -gerbes over Lie groupoids.