# Universal property of the homotopy limit/colimit.

I have been trying to find a reference for what I have heard is a universal property which defines homotopy limits and colimits.

In the category Top, colimits can be defined using the following universal property:

Let $$\mathcal{F}:\mathcal{D} \rightarrow \textbf{Top}$$ be a diagram of shape $$\mathcal{D}$$ in Top. The colimit of $$\mathcal{F}$$ is defined to be an object $$O$$ together with maps $$\phi_d$$ for every object $$d \in \mathcal{D}$$ (such that everything commutes) such that if there is another co-cone under the diagram with tip $$Z$$ (and maps $${\psi_d}$$ such that everything commutes), then there is a unique map $$\Phi:O \rightarrow Z$$ making everything commute.

There is a similar universal property for the limit of $$\mathcal{F}$$.

I'm hoping to find a universal property along these lines which the homotopy colimit satisfies. What I initially thought was that we could simply use the universal property which the colimit satisfies, except change the category from Top to HoTop (where Hotop is the category with spaces for objects and homotopy classes of maps for morphisms).

This doesn't seem to be satisfied by what I know to be examples of the homotopy colimit of a diagram, though.

For example, given the diagram which sends $$S^1$$ into two distinct copies of the disk $$D^2$$, the homotopy colimit "is" $$S^2$$. Then there are at least two distinct morphisms in HoTop $$S^2 \rightarrow S^2$$ (i.e. two distinct homotopy classes of maps) which make everything commute up to homotopy.

Is there (alternatively) a universal property of the homotopy colimit which says:

Let $$\mathcal{F}:\mathcal{D} \rightarrow \textbf{HoTop}$$ be a diagram of shape $$\mathcal{D}$$ in HoTop. The colimit of $$\mathcal{F}$$ is defined to be an object $$O$$ together with morphisms $$\phi_d$$ for every object $$d \in \mathcal{D}$$ (such that everything commutes) such that if there is another co-cone under the diagram with tip $$Z$$ (and morphisms $${\psi_d}$$ such that everything commutes), then there is a (not necessarily unique) morphism $$\Phi:O \rightarrow Z$$ making everything commute.

Commutativity in the homotopy category being homotopy commutativity in Top, and morphisms being homotopy classes of morphisms in Top

If not, is there anything like that? Is the simplest bet otherwise to go with "an object which is homotopy equivalent to a specific construction of the homotopy colimit"?

Under a suitable framework, homotopy colimits and limits satisfy a "local homotopical universal property". For example, homotopy colimits represent "homotopy coherent cones". See section 10 of Shulman's "Homotopy limits and colimits and enriched homotopy theory" for the definitions and a discussion. Riehl's Categorical homotopy theory also discusses this in section 7.7.

Homotopy colimits and limits also have a global definition in terms of derived functors when they make sense. So this could also serve as an universal property defining them.

• And so in this case, could one use the universal property (for colimits) which I state above? I.E. The homotopy colimit of a diagram $\mathcal{F}$ is a co-cone $Z$ where for any other co-cone $\overline{Z}$ there exists a homotopy coherent map $f:Z \rightarrow \overline{Z}$?
– Matt
Commented Feb 17, 2019 at 12:44
• I don't think so. Your proposed universal property only asks for homotopy commutativity, but homotopy coherence is much more structure. In fact, your definition sort of looks like a colimit in the homotopy category, but in general those often fail to exist.
– JHF
Commented Feb 17, 2019 at 22:23