Here I just include some things I learned that I did not read in books that I think are valuable lessons. Some of these might be obvious, but I feel like they took me a long time to digest. I learned most of these from my advisors, professors, and colleagues.

#### Structure and property

A given mathematical object can be endowed with additional structure. This structure is additional data. For instance, a vector space can be endowed with an inner product. A property is a feature of a mathematical object. For instance, every vector space has a basis. A *choice* of a basis is structure, but the existence is a property.

#### Definition vs. construction

A definition of a mathematical object need not assume its existence. A construction provides a proof for the existence of some mathematical object. Sometimes, a property may characterize a mathematical object or a class of mathematical objects. In this case, a definition of a mathematical object can be defined in terms of its characterizing properties. For example, a *definition* of a tensor product of vector spaces is given by its universal property. A *construction* of a tensor product of vector spaces is given in terms of generators and relations. As another example, entropy is typically written down as a formula. However, a proper definition of entropy is some function satisfying certain (physically reasonable) postulates. The formula is a particular construction of this definition and in certain cases it is unique (up to a choice of units).

#### Invariants are functors

What are invariants? Given a mathematical structure, the collection of these mathematical structures and all of their relations (morphisms and so on) form some sort of a category. *Invariants* are *functors* from this category to any other category. Homotopy theory provides another interesting perspective on invariants. In homotopy theory, one is interested in invariantsĀ *under homotopy*. It turns out there is a *universal invariant* in the sense that *every other *invariant factors through this one. More precisely, let $latex C$ be a model category or a category with weak equivalences. Let $latex \mathrm{ho}C$ be the associated homotopy category obtained by localizing $latex C$ with respect to the weak equivalences. Then the localization $latex L : C \to \mathrm{ho}C$ satisfies the property that for any other functor $latex F: C \to D$ that sends weak equivalences to isomorphisms, there exists a unique functor $latex \mathrm{ho} C \to D$ making the obvious diagram commute. I first learned this from Dan Freed and Mike Hopkins in their article “Chern-Weil Forms and Abstract Homotopy Theory.”