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Stream: learning: questions

Topic: Pseudopushouts and pseudocoequalisers of categories

Emily (Aug 10 2021 at 11:08):

Lately I've been trying to apply the definitions of free loop space objects, loop space objects, and suspension objects to the $2$-category of groupoids. These are defined in the nLab as homotopy co/limits, which in the model category of groupoids are the same as pseudo-co/limits, by a result of Gambino. Moreover, since Grpd is reflective in Cats, it follows that the inclusion $\mathsf{Grpd}\hookrightarrow\mathsf{Cats}$ creates pseudo-co/limits, and thus we can just compute those as pseudo-co/limits in Cats. So then I turned to look for descriptions of some pseudo-co/limits of categories, among which I found those of pseudoequalisers and pseudopullbacks. From these two descriptions, I've seen that:
1) The "free loop space" of a groupoid $\mathcal{G}$ is the groupoid whose objects are quadruples $(A,B,f,g)$ with $A$ and $B$ objects of $\mathcal{G}$ and $f,g\colon A\rightrightarrows B$ morphisms of $\mathcal{G}$, and whose morphisms are pairs of morphisms $A\to A'$ and $B\to B'$ making the natural square commute;
2) The "based loop space" of a pointed groupoid $(\mathcal{G},x_0)$ is given by $\mathrm{Aut}(x_0)_\mathsf{disc}\overset{\mathrm{def}}{=}\mathrm{Hom}_{\mathcal{G}}(x,x)_\mathsf{disc}$, the discrete groupoid associated to the automorphism group of $x_0$ in $\mathcal{G}$.

On the other hand, I've found it difficult to find descriptions of pseudocoequalisers and pseudopushouts of categories in the literature (I've also failed to figure out what these are, as I'm not very familiar with pseudo-co/limits yet). So I have been thinking about the following questions:

• How are pseudopushouts and pseudocoequalisers of categories explicitly constructed?
• More restrictively, assuming the above is too hard/bothersome (or as nice examples), what are

  1. The pseudocoequaliser of the pair $\mathsf{pt}\rightrightarrows\mathsf{pt}$ (which is supposed to be the analogue of $S^1$ in groupoids)?
  2. The pseudopushout $\Sigma{\mathcal{G}}\overset{\mathrm{def}}{=}\mathsf{pt}\coprod^{\mathsf{ps}}_{\mathcal{G}}\mathsf{pt}$, with $\mathcal{G}$ a groupoid and $\mathsf{pt}$ the punctual category (supposed to be the suspension of $\mathcal{G}$)?

Sam Speight (Aug 10 2021 at 11:37):

Towards your first question, coequalizers of categories are described explicitly using "generalized congruences" in a paper by Bednarczyk, Borzyszkowski and Powlowski.

Sam Speight (Aug 10 2021 at 11:45):

I have extended this to strict 2-categories, and have an idea for how you can weaken this for bicategories and pseudo-stuff.

Zhen Lin Low (Aug 10 2021 at 12:44):

You can construct them using strict colimits by using the same kinds of explicit homotopy colimit constructions used classically. There's no easy explicit description since they involve adjoining isomorphisms.

Reid Barton (Aug 10 2021 at 12:51):

You can also identify groupoids (up to equivalence) with 1-truncated spaces and then use literally the same constructions as in the topological space, except that to form colimits you need to 1-truncate the result.

Reid Barton (Aug 10 2021 at 12:52):

The answers are 1. $B\mathbb{Z}$, 2. the same as the suspension of $\pi_0 G$ (the set of isomorphism classes of objects of $G$).

Zhen Lin Low (Aug 10 2021 at 13:07):

Indeed, though maybe it's worth pointing out that there's also a notion of pseudocolimit that is definite up to isomorphism, rather than equivalence. In this case the pseudocoequaliser in question has two objects and the pseudopushout has two more objects than $\mathcal{G}$ has.

Emily (Aug 10 2021 at 16:29):

Thanks, @Sam Speight, @Zhen Lin Low, and @Reid Barton!

Emily (Aug 10 2021 at 16:30):

Zhen Lin Low said:

You can construct them using strict colimits by using the same kinds of explicit homotopy colimit constructions used classically. There's no easy explicit description since they involve adjoining isomorphisms.

Could you please elaborate a bit? In particular, what kind of strict colimit do we get by doing this to pseudopushouts?

Emily (Aug 10 2021 at 16:30):

Reid Barton said:

You can also identify groupoids (up to equivalence) with 1-truncated spaces and then use literally the same constructions as in the topological space, except that to form colimits you need to 1-truncate the result.

Is it correct to say then that the colimit of a functor $F\colon\mathcal{K}\to\mathsf{Grpd}$ is given by $\mathrm{colim}(F)\cong\Pi_{1}(\mathrm{colim}({|F|})))$, with $|F|\colon\mathcal{K}\to\mathsf{Top}$ the composition of $F$ with the classifying space functor?

Zhen Lin Low (Aug 11 2021 at 02:13):

To construct the pseudopushout of $B \leftarrow A \rightarrow C$, take the ordinary colimit of $B \leftarrow A \times \{ 1 \} \rightarrow A \times I \leftarrow A \times \{ 0 \} \rightarrow A \times I \leftarrow A \times \{ 1 \} \rightarrow C$, where $I$ is the groupoid $\{ 0 \cong 1 \}$.
In words, this is the groupoid obtained by taking the disjoint union of $A, B, C$ and then adjoining isomorphisms according to $A \to B$ and $A \to C$.

Emily (Aug 13 2021 at 03:38):

Zhen Lin Low said:

To construct the pseudopushout of $B \leftarrow A \rightarrow C$, take the ordinary colimit of $B \leftarrow A \times \{ 1 \} \rightarrow A \times I \leftarrow A \times \{ 0 \} \rightarrow A \times I \leftarrow A \times \{ 1 \} \rightarrow C$, where $I$ is the groupoid $\{ 0 \cong 1 \}$.
In words, this is the groupoid obtained by taking the disjoint union of $A, B, C$ and then adjoining isomorphisms according to $A \to B$ and $A \to C$.

Oh I see! Thanks, Zhen!