Here is an example that surprised me at some time in the past. Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs. Here a directed graph is simply a 4-tuple (V,E,s,t) where an arc e∈E starts at s(e)∈V and ends at t(e)∈V. Fibrations are maps that induce a surjection on the set of outgoing arcs of each vertex, cofibrations are embeddings obtained by attaching a bunch of trees, and weak equivalences are maps that induce a bijection on the sets of cycles.

In this model structure fibrant objects are graphs without sinks and cofibrant objects are graphs with exactly one incoming arc for every vertex. Cofibrant replacement replaces a graph by the disjoint union of its cycles with the obvious morphism into the original graph.

We have a chain of inclusions of categories A→B→C→D, where D is the topos of directed graphs, C is the full subcategory of D consisting of all graphs with exactly one incoming arc for each vertex, B is the full subcategory of C consisting of all graphs with exactly one outgoing arc for each vertex, and A is the full subcategory of B consisting of all graphs such that s=t.

Each functor is a part of a Quillen adjunction and total left and right derived functors compute nontrivial information about graphs under consideration.

Two finite graphs are homotopy equivalent iff they are isospectral iff their zeta-functions coincide.

Here is an example that surprised me at some time in the past. Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs. Here a directed graph is simply a $4$-tuple $(V,E,s,t)$ where an arc $e \in E$ starts at $s(e) \in V$ and ends at $t(e) \in V$. Fibrations are maps that induce a surjection on the set of outgoing arcs of each vertex, cofibrations are embeddings obtained by attaching a bunch of trees, and weak equivalences are maps that induce a bijection on the sets of cycles.

In this model structure fibrant objects are graphs without sinks and cofibrant objects are graphs with exactly one incoming arc for every vertex. Cofibrant replacement replaces a graph by the disjoint union of its cycles with the obvious morphism into the original graph.

We have a chain of inclusions of categories $A\to B \to C\to D$, where $D$ is the topos of directed graphs, $C$ is the full subcategory of $D$ consisting of all graphs with exactly one incoming arc for each vertex, $B$ is the full subcategory of $C$ consisting of all graphs with exactly one outgoing arc for each vertex, and $A$ is the full subcategory of $B$ consisting of all graphs such that $s=t$.

Each functor is a part of a Quillen adjunction and total left and right derived functors compute nontrivial information about graphs under consideration.

Two finite graphs are homotopy equivalent iff they are isospectral iff their zeta-functions coincide.

Here is an example that surprised me at some time in the past. Bisson and Tserno introduce a nontrivial model structure on the topos of directed graphs. Here a directed graph is simply a 4-tuple (V,E,s,t) where an arc e∈E starts at s(e)∈V and ends at t(e)∈V. Fibrations are maps that induce a surjection on the set of outgoing arcs of each vertex, cofibrations are embeddings obtained by attaching a bunch of trees, and weak equivalences are maps that induce a bijection on the sets of cycles.

In this model structure fibrant objects are graphs without sinks and cofibrant objects are graphs with exactly one incoming arc for every vertex. Cofibrant replacement replaces a graph by the disjoint union of its cycles with the obvious morphism into the original graph.

We have a chain of inclusions of categories A→B→C→D, where D is the topos of directed graphs, C is the full subcategory of D consisting of all graphs with exactly one incoming arc for each vertex, B is the full subcategory of C consisting of all graphs with exactly one outgoing arc for each vertex, and A is the full subcategory of B consisting of all graphs such that s=t.

Each functor is a part of a Quillen adjunction and total left and right derived functors compute nontrivial information about graphs under consideration.

Two finite graphs are homotopy equivalent iff they are isospectral iff their zeta-functions coincide.

Here is an example that surprised me at some time in the past. Bisson and Tsemo introduce a nontrivial model structure on the topos of directed graphs. Here a directed graph is simply a 4-tuple (V,E,s,t) where an arc e∈E starts at s(e)∈V and ends at t(e)∈V. Fibrations are maps that induce a surjection on the set of outgoing arcs of each vertex, cofibrations are embeddings obtained by attaching a bunch of trees, and weak equivalences are maps that induce a bijection on the sets of cycles.

In this model structure fibrant objects are graphs without sinks and cofibrant objects are graphs with exactly one incoming arc for every vertex. Cofibrant replacement replaces a graph by the disjoint union of its cycles with the obvious morphism into the original graph.

We have a chain of inclusions of categories A→B→C→D, where D is the topos of directed graphs, C is the full subcategory of D consisting of all graphs with exactly one incoming arc for each vertex, B is the full subcategory of C consisting of all graphs with exactly one outgoing arc for each vertex, and A is the full subcategory of B consisting of all graphs such that s=t.

Each functor is a part of a Quillen adjunction and total left and right derived functors compute nontrivial information about graphs under consideration.

Two finite graphs are homotopy equivalent iff they are isospectral iff their zeta-functions coincide.