Homotopy Colimits

So, if you've read much about category theory, you've encountered the dual notions of the categorical limit and colimit. For a refresher, here's the wikipedia page. If you've taken much topology, you've heard about homotopies. So here's a mathematical mashup for you: homotopy colimits.

What's wrong with just using colimits? It all depends on your context. In the context of homotopy theory, if you've got two functors to spaces, $F$ and $G$, out of the same category, say $\mathbf{C}$, and for every $c \in \mathbf{C}$ you have a homotopy equivalence $F(c)\simeq G(c)$, then you'd probably wish for $\mathrm{colim}F\simeq \mathrm{colim}G$. Maybe you're a bit more careful, and would only expect this to hold when there is a natural transformation $\eta:F\rightarrow G$ such that $\etc(c):F(c)\rightarrow G(c)$ is a homotopy equivalence for all $c\in \mathbf{C}$. But still, this is not enough.

Consider the following example. Take $\mathbf{C}$ to be what you might call the pushout category: $1\leftarrow 12\rightarrow 2$. To define $F$ and $G$, let's set up some more notation. Recall that $D^2$ represents the standard 2-dimensional closed disk, and it's boundary is the circle, $S^1$. Let $*$ denote the one-point space. Now define $F$ via the diagram $D^2\leftarrow S^1\rightarrow D^2$, where both maps are the standard inclusion, and $G$ via $*\leftarrow S^1\rightarrow *$. There is an obvious map of diagrams (natural transformation) from $F$ to $G$, and on each object $c$ of $\mathbf{C}$ the map $F(c)\rightarrow G(c)$ is an equivalence. Then $\mathrm{colim}F$ is the space obtained by joining two disks along their boundary, i.e., the sphere $S^2$. On the other hand $\mathrm{colim}G$ is simply the one-point space $*$. Since $S^2$ is not contractible... there's something unsuitable about the colimit from a homotopy viewpoint. So how do you fix it?

Let's go back and think about what the standard colimit is (with our diagram category $\mathbf{C}$ being suitably small - I generally visualize them as a finite set of objects with a finite set of arrows). In this situation, the standard colimit, $\mathrm{colim}F$, is the quotient space of $\coprod_{c\in \mathbf{C}}F(c)$ where $x\in F(c)$ is identified with $y\in F(d)$ if there is an arrow $f:c\rightarrow d$ in $\mathbf{C}$ so that $F(f):F(c)\rightarrow F(d)$ has $F(f)(x)=y$. That is, we glue $F(c)$ to $F(d)$ using maps $F(f)$ when $f:c\rightarrow d$.

One example you might think about is when $\mathbf{C}$ is just $1\rightarrow 2$. In this case, the colimit of any functor $F$ is just $F(2)$ (2 is, after all, a final object). So all of the information about $F(1)$ gets completely lost. That feels dangerous.

Let's see what we can do to remedy the situation. To begin, lets remain in the case where $\mathbf{C}$ is the simple category $1\rightarrow 2$ (call the non-identity arrow $f$). Instead of just gluing $F(1)$ to $F(2)$ and losing $F(1)$, consider the mapping cylinder. This is the quotient of the disjoint union of $F(1)\times I$ and $F(2)$ ($I$ here is the unit interval $[0,1]$), obtained by gluing together $(x,1)\in F(1)\times I$ with $F(f)(x)\in F(2)$. You may notice that the homotopy type of $\mathrm{colim}F$ is the same as the homotopy type of the mapping cylinder (you think about just projecting $F(1)\times I$ into $F(1)\times \{1\}\subset F(2)$), but they are not homeomorphic. We still have a copy of $F(1)$ sitting inside the mapping cylinder, as $F(1)\times \{0\}$. I generally visualize this as one end of a tailpipe, with the other end stuck to the space $F(2)$.

It turns out that this is the 'right' construction of the colimit for homotopy theory (in this example), and is given the name 'homotopy colimit' ($\mathrm{hocolim}$). What about other examples? What about the pushout category we started with ($1\leftarrow 12\rightarrow 2$)? The way to generalize the constrution of the homotopy colimit to that situation is fairly similar. Instead of using just one mapping cylinder, the best way to think about the homotopy colimit over the pushout category is as two mapping cylinder (one for $1\leftarrow 12$ and one for $12\rightarrow 2$), joined together on their end copies of $F(12)$. Go back to our motivating example, and see if you get the same homotopy type for $\mathrm{hocolim}F$ and $\mathrm{hocolim}G$, using this double mapping cylinder. What'd you get?

How about even more general diagrams? Well, for that you need another couple of notions: the nerve of a category, and the geometric realization of a category (which is the geometric realization of its nerve, a simplicial set). You can read about undercategories (on that page, it is the example of 'Category of objects under A') and nerves on wikipedia (well, lots of places, I'm sure).

For an object $c\in \mathbf{C}$, let me denote by $c\downarrow \mathbf{C}$ the 'category of objects under $c$', and denote the geometric realization of (the nerve of) that category by $|c\downarrow \mathbf{C}|$. You may notice that $id:c\rightarrow c$ is an initial object in $c\downarrow \mathbf{C}$, and this means that $|c\downarrow \mathbf{C}|$ is contractible, for all $c$. Momentarily we will use $F(c)\times |c\downarrow\mathbf{C}|$, but I would like to point out first that we may consider $F(c)$ as a subspace via $F(c)\times id$. Also, since $|c\downarrow \mathbf{C}|$ is contractible, $F(c)\times |c\downarrow \mathbf{C}|$ is homotopy equivalent to $F(c)$.

The general construction for $\mathrm{hocolim}F$ is then defined as a quotient space of $\coprod_{c\in \mathbf{C}}F(c)\times |c\downarrow\mathbf{C}|$. Points of the form $(x,f)$, where $x\in F(c)$ and $f:c\rightarrow d\in c\downarrow \mathbf{C}$ (technically, you should take $f$ as the vertex in the geometric realization...) then get identified to $F(f)(x)\in F(d)$, as before (I think some more things have to get identified also, but I'm getting sleepy, so it's not coming to me). Here we are using the inclusion $F(d)\subset F(d)\times |d\downarrow \mathbf{C}|$, mentioned in the previous paragraph.

It is, at this point, probably a good idea to go back to the two examples above, the mapping cylinder and double mapping cylinder, and convince yourself that the general construction above is really a generalization of these cases.

One thing you may notice is that the maps $|c\downarrow \mathbf{C}|\rightarrow *$ (which are homotopy equivalences, remember) induce maps $F(c)\times |c\downarrow \mathbf{C}|\rightarrow F(c)$. The homotopy colimit construction is built from the domains of these arrows, while the standard colimit is built from the codomains of these arrows. When it all boils down, it turns out that you get a map $\mathrm{hocolim}F\rightarrow \mathrm{colim}F$. I always think this is interesting because you expect maps to come out of colimits, not go in. You might wonder when this map is an equivalence. I'll let you.

I'd like to give you a few more examples, while I'm here. Both examples are diagrams indexed on the pushout category above, and such homotopy colimits are frequently called 'homotopy pushouts'.

1. The homotopy pushout of the diagram $*\leftarrow X\rightarrow *$ is the suspension of $X$ (so you can check your answer to our motivating example above).
   
2. The homotopy pushout of the diagram $X\leftarrow X\times Y\rightarrow Y$ (both maps the canonical projections) is the join, $X\star Y$.
   

If you're just itching to know about the dual notion of homotopy limits... well... undercategories change to overcategories, and quotients turn into mapping spaces. Briefly, you get a space of natural transformations: $\mathrm{Nat}_{\mathbf{C}}(|\mathbf{C}\downarrow -|,F(-))$.

So anyway, there's that. If you notice any errors, or you've got more fun examples, please leave a comment. I wasn't too formal about some things, but hopefully I've got it mostly ok.