Articles of higher category theory

Reasons for coherence for bi/monoidal categories

Here by coherence conditions I mean those axioms imposed on associators and unities that grant that the groupoid generated by such morphisms is a poset, i.e. any two parallels morphisms in this groupoid are equal. These condition implies that such weak structures are equivalent (in a suitable sense) to strict ones. That said, where does […]

When should one learn about $(\infty,1)$-categories?

I’ve been doing a lot of reading on homotopy theory. I’m very drawn to this subject as it seems to unify a lot of topology under simple principles. The problem seems to be that the deeper I go the more confused I get since the number of possible formalisms grows exponentially. The more i read […]

Alternative categorification of metric spaces?

This is a more articulate version of a question I asked a few days ago. I have made an attempt to “categorify” the theory of metric spaces. Informally, in the spirit of representing a topological space as a category (as in topos theory), I think I have a way to make a 2-category that encodes […]

Is it possible to formalize (higher) category theory as a one-sorted theory, just like we did with set theory?

Set theory is typically formalized as a one-sorted theory without urelements. Is it possible to do the same with category theory or higher category theory, formalizing the whole affair as a theory with only one type of object? I know that this runs counter to what some feel is the philosophical spirit of category theory, […]

What is a copresheaf on a “precategory”?

Let $\mathcal{S}$ be a category with pullbacks. A precategory $\mathbb{C}$ in the sense of Borceux and Janelidze [Galois Theories, §7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and morphisms $d_2^0, d_2^1, d_2^2 : C_2 \to C_1$, $d_1^0, d_1^1 : C_1 \to C_0$, $s_0^0 : C_0 \to C_1$, satisfying the following fragment of the simplicial […]

Pullbacks of categories

Let $\mathfrak{Cat}$ be the 2-category of small categories, functors, and natural transformations. Consider the following diagram in $\mathfrak{Cat}$: $$\mathbb{D} \stackrel{F}{\longrightarrow} \mathbb{C} \stackrel{G}{\longleftarrow} \mathbb{E}$$ There are several notions of pullback one could investigate in $\mathfrak{Cat}$: The ordinary pullback in the underlying 1-category $\textbf{Cat}$: these exist and are unique, by ordinary abstract nonsense. Explicitly, $\mathbb{D} \mathbin{\stackrel{1}{\times}_\mathbb{C}} \mathbb{E}$ […]

Can Spectra be described as abelian group objects in the category of Spaces? (in some appropriate $\infty$-sense)

I’m not a topologist and I’m trying to understand a little bit about spectra. I’ve been told that spectra are the homotopical version of abelian groups. Can anyone expand on this point? Apparently it should be somewhere in Lurie’s work, but all I could find (thanks to moritz groth’s notes) is that Spectra are the […]

Concrete examples of 2-categories

I’ve been reading some of John Baez’s work on 2-categories (eg here) and have been trying to visualize some of the constructions he gives. I’m interested in coming up with ‘concrete’ examples of 2-categories. As an example of what I don’t mean, I know that the category Cat forms a 2-category, where the objects are […]