Homotopy Pullbacks and Homotopy Groups in CAT

In their work on algebraic quantum field theory, Roberts, Ruzzi, and Vasselli gave a purely combinatorial description for the exact sequence of fundamental groups associated to certain fibered functors between posets (poset net bundles). This relies on a description of homotopy in terms of zig-zags of morphisms, similar to work of Evrard from the 1970s.

The work discussed in this talk grew out of a desire to give a combinatorial description for the entire long exact sequence in homotopy. To this end, we describe a functor from cospans in $\underset{¯}{\mathrm{Cat}}$ to $\underset{¯}{\mathrm{Cat}}$, which models the homotopy pullback after passing to geometric realizations. This functor is a categorical version of the usual description of homotopy pullbacks in $\underset{¯}{\mathrm{Top}}$, but with geometric paths replaced by zig-zags of morphisms (of arbitrary length). As such, our result for homotopy pullbacks is an infinitary version of Barwick–Kan’s Theorem B_n. In the special case of homotopy fibers, we recover a result of Shoikhet (simplifying an earlier model for homotopy fibers due to Evrard); this model is an infinitary version of Dwyer–Hirschhorn–Kan’s Theorem B_n (itself a variation on Quillen’s Theorem B).

Applying our model for homotopy pullbacks to a cospan

$$\{C\}\hookrightarrow 𝒞\hookleftarrow \{C\},$$

we recover the categorical model for based loop spaces introduced by Evrard, from which one can extract Evrard’s model for homotopy groups in terms of $n$–dimensional zig-zag diagrams in the underlying category. Our approach allows for a combinatorial description of the boundary map in the long-exact Mayer–Vietoris sequence associated to an arbitrary homotopy pullback; in the setting of algebraic quantum field theory, this yields the desired combinatorial description for the long exact sequence of a poset net bundle.