# Notes for Florian

### Abstract
In this note, you'll find a little bit of the history of homotopy theory - together with links to some good references - which help to provide context for infinity categories. Infinity categories developed simultaneously in category theory and homotopy theory really at the intersection of the fields. But there does not seem to be an expository paper available that really explains what problem this invention was trying to solve, at least from the perspective of homotopy theory (infinity categories also arose out of higher category theory, which had its own independent reasons for being).

### Homotopy Limits and Colimits

In a category, the (co)limit is a universal (co)cone for a diagram. This definition is well understood, see for example [Robin's course notes](https://pages.cpsc.ucalgary.ca/%7Erobin/class/617/notes.pdf) or any basic category theory textbook. However, this definition is insufficient in topology because it does not preserve homotpy equivalence. The standard example is to take the pushout:
$$ \bullet \leftarrow S^1 \rightarrow \bullet $$
The strict pushout is of course the one-point space $\bullet$. But this diagram is "homotopic to"
$$ D^1 \leftarrow S^1 \rightarrow D^1 $$
(where the maps are given by the inclusion of the boundary) and the strict pushout of this diagram is $S^2$. Since $S^2 \not\simeq \bullet$, this tells us that the pushout does not preserve homotopy. Topologists acknowledged limits and colimits, but rejected them as the "correct" notion of limits and colimits in topology for this reason. A good account of the problem and the proposed solution can be found in Munson-Volic's [Cubical Homotopy Theory](https://palmer.wellesley.edu/~ivolic/pdf/Papers/CubicalHomotopyTheory.pdf), Chapter 3. What you will find there is the *classical* approach to constructing a homotopy limit/colimit which is homotopy invariant. But this construction is categorically non-sensical - the resulting construction does not satisfy a universal property. Quillen made sense of this in the 1960's with the advent of [Homotopical Algebra](https://link.springer.com/book/10.1007/BFb0097438). In a Quillen Model Category, there are two classes of maps, fibrations and cofibrations. A third class of maps, the weak equivalences, correspond to homotopy equivalences. Fibrations behave well with respect to limits and cofibrations behave well with respect to colimits. Any morphism $f:A\to B$ can be factored as $f=g\circ h$ where $h$ is a cofibraion and weak equivalence, and $g$ is a fibration. The morphism $f$ can also be factored as $f=g'\circ h'$ where $g\$ is a cofibration, and $f'$ is a fibration and a weak equivalence. In topology, the cofibrations correspond to inclusion maps (it's slightly more complicated - they are inclusions if your spaces are all CW complexes). The fibrations are a slight generalization of covering maps. The map
$$S^1\rightarrow \bullet$$
factors as
$$ S^1 \rightarrow D^2 \rightarrow \bullet$$
and the first of these is a cofibration, while the second is a fibration and homotopy equivalence.

The topologist's homotopy colimit construction was understood by Quillen as: use the existence of factorizations to take an arbitrary diagram and replace it with an equivalent one in the category whose morphisms are all cofibrations. Then take the strict colimit. This does result in a sort of universal property, but it is quite awkward to state this in any precise way. This presents our first challenge: what *is* the universal property defining a homotopy (co)limit?

### Homotopy coherence vs. homotopy commutativity

Homotopy pushouts and pullbacks tend to produce homotopy commutative diagrams. For example, the homotopy pushout of
$$ \bullet \leftarrow S^1 \rightarrow \bullet$$
produces a square diagram (which HackMD cant' tex)
whose final vertex is $S^2$. The two maps $\bullet \to S^2$ in this diagram correspond to the inclusion of the north and south pole of $S^2$, respectively. The resulting diagram does not commute strictly, since one way of going around the square produces
$$ S^1 \to \bullet \to S^2$$
where the image of $S^1$ is the north pole. The other way of going around the square produces
$$ S^1 \to \bullet \to S^2$$
where the image of $S^1$ is the south pole. Although these two maps are not strictly equal, they are homotopic, basically because there is a path from the north to the south pole in $S^2$. This example illustrates the idea of *homotopy commutativity*. In practice, homotopy theorists would treat this diagram as if it were commutative. Occassionally, the homotopy between the two maps must be remembered and kept track of - but this information is often surpressed. Keeping track of this homotopy would mean that we are keeping track of *coherence*. In this case, we would think of this diagram as being *homotopy coherent* instead of *homotopy commutative*. The difference is that in the second case, we only acknowledge the existence of a homotopy which makes teh diagram commute. In the first case, we have a specific homotopy in mind.

Lurie himself identifies this idea - the tendency in homotopy theory to deal with homotopy commutative diagrams while surpressing the homotopy - as the problem that infinity categories solved. He identifies infinity categories as a way of solving this problem by moving from homotopy commutative diagrams to homotopy coherent diagrams. For an account of this, see the first chapter of [Higher Topos Theory](https://people.math.harvard.edu/~lurie/papers/highertopoi.pdf). I gave more specific references in teams.

This also has implications for the existence of homotopy limits and colimits. Now, it is possible to construct a homotopy (co)limit as a strict (co)limit in an infinity category together with a coherence (contained in higher cells) which correctly interpret the resulting diagram as a homotopy commutative diagram. So although Lurie does not state the trouble with homotopy (co)limits as the motivating problem for infinity categories, this problem is closely related to the one he *does* cite - i.e. homotopy coherence.

### Infinity Category Resources

I see that there is an infinity category theory course happening at Rochester, given by someone whose perspective aligns with my own. You can try browsing the [lecture notes by A Mattoo](https://www.math.columbia.edu/~amattoo/S22_Infinity_Categories_Talks.pdf).