Reconstruction of graded groupoids from graded Steinberg algebras

We show how to reconstruct a graded ample Hausdorff groupoid with topologically principal neutrally-graded component from the ring structure of its graded Steinberg algebra over any commutative integral domain with 1, together with the embedding of the canonical abelian subring of functions supported on the unit space. We deduce that diagonal-preserving ring isomorphism of Leavitt path algebras implies $C^*$-isomorphism of $C^*$-algebras for graphs $E$ and $F$ in which every cycle has an exit. This is a joint work with Joan Bosa, Roozbeh Hazrat and Aidan Sims.Universidad de Málaga. Campus de Excelencia internacional Andalucía Tec

Strict \infty-groupoids are Grothendieck \infty-groupoids

We show that there exists a canonical functor from the category of strict \infty-groupoids to the category of Grothendieck \infty-groupoids and that this functor is fully faithful. As a main ingredient, we prove that free strict \infty-groupoids on a globular pasting scheme are weakly contractible.Comment: 22 pages, v2: revised according to referee's comments, in particular: new organization of the pape

The Brown-Golasinski model structure on strict ∞\infty-groupoids revisited

We prove that the folk model structure on strict $\infty$-categories transfers to the category of strict $\infty$-groupoids (and more generally to the category of strict $(\infty, n)$-categories), and that the resulting model structure on strict $\infty$-groupoids coincides with the one defined by Brown and Golasinski via crossed complexes.Comment: 24 pages, v2: generalization to strict $(\infty, n)$-categories adde

Tensor products of Leavitt path algebras

We compute the Hochschild homology of Leavitt path algebras over a field $k$. As an application, we show that $L_2$ and $L_2\otimes L_2$ have different Hochschild homologies, and so they are not Morita equivalent; in particular they are not isomorphic. Similarly, $L_\infty$ and $L_\infty\otimes L_\infty$ are distinguished by their Hochschild homologies and so they are not Morita equivalent either. By contrast, we show that $K$-theory cannot distinguish these algebras; we have $K_*(L_2)=K_*(L_2\otimes L_2)=0$ and $K_*(L_\infty)=K_*(L_\infty\otimes L_\infty)=K_*(k)$.Comment: 10 pages. Added hypothesis to Corolary 4.5; Example 5.2 expanded, other cosmetic changes, including an e-mail address and some dashes. Final version, to appear in PAM