Many people working in graph theory today are interested in thinking of graphs as analogous to spaces, and developing homotopy theories for graphs. One particular categorical point of view on this idea can be found in the work of Lawvere, dating back to the 1980s. Lawvere observed that some elementary toposes seem to deserve to be thought of as…

Algebraic theories, such as the theories of groups, rings, and
modules over a given ring, are often studied category theoretically
as finitary monads on Set. Which such theories have the
property that all their algebras are free? It turns out that we can
write them all down relatively easily! Proving this result, however, is
…

A 2-group is a groupoid with a structure analogous to that of a group. By a representation of a 2-group G I mean a (weak) 2-functor from G[1], the one-object 2-groupoid with G as 2-group of self-equivalences of the unique object, to some target 2-category C. Taking the 2-category of Baez-Crans 2-vector spaces as C leads to a bad…