Questions tagged [higher-category-theory]

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. (Def: http://en.m.wikipedia.org/wiki/Higher_category_theory)

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Reference: Wikipedia.

Topics (between others): Strict higher categories, Weak higher categories, Quasi-categories.

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Textbooks on higher category theory

What textbooks on higher category theory are there? What books do you recommend? I am looking for self-contained introductions, no research reports. There are lots of informal summaries and arXiv papers, but I am really only asking for textbooks…
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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…
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Sheaves of categories

I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a topological space $X$, which leads one to the notion of…
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What is an intuitive Geometrical explanation of a "sheaf?"

As I understand it, a sheaf is a very broad concept, but is most often used when referencing a function that maps algebraic structures (like rings) to points on a manifolds. Is a sheaf just yet another type of function, but built specifically for…
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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}…
Zhen Lin
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How to construct a quasi-category from a category with weak equivalences?

Let $(\mathcal{C},W)$ be a pair with $\mathcal{C}$ a category and $W$ a wide (containing all objects) subcategory. Such a pair represents an $(\infty,1)$-category. One model for such gadgets is a quasi-category (a simplicial set satisfying the weak…
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Why is it that homotopy is better described by weak equivalences than by homotopies?

I've been reading about (abstract or not) homotopy theory, and I seem to have understood (correct me if I'm wrong) that weak equivalences describe homotopy better than homotopies, in the following sense : Intuitively, if I wanted to abstract away…
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What's a double category with one object?

Categories with one object are equivalent to monoids. $2$-categories with one object are equivalent to monoidal categories. Therefore, I am wondering whether double categories with one object are equivalent to some known or interesting algebraic…
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Can you explain Lawvere's work on Hegel to someone who knows basic category theory?

I have worked my way through Simmons' intro to category theory. I also know what a subobject classifier/elementary topos is. Is there anyone who could explain what Lawvere did with Hegel's work (described here), as well as what Hegel's ideas are in…
user52969
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The $2$-category of monoids

People sometimes say that monoids are "categories with one object". In fact people sometimes suggest that this is the natural definition of a monoid (and likewise "groupoid with one object" as the definition of a group). But categories naturally…
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Why is the 'mapping space' between two objects in a quasi-category a Kan complex?

Let $X$ be a quasi-category. (i.e. a simplicial set satisfying the weak Kan extension condition). Given two 'objects' $a,b\in X_0$ in $X$, one defines the mapping space $X(a,b)$ to be the pullback of $$ X^{\Delta^1}\to X^{\partial\Delta^1}\cong…
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Higher categories for category theorists?

I'm not actually a category theorist, but assume I have some background in category theory and am pretty comfortable with it; and I want to learn higher category theory (say in the sense of Boardmann-Vogt-Joyal-Lurie) but don't want to have to delve…
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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…
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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…
Jacob Bell
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Difference between $[pt/G]$ and $BG$

Let $G$ be a finite group. In topological category, we have the quotient stack of a point by $G$, denoted by $[pt/G]$. We also have the classifying space $BG$, which is a topological space. I am a bit confused as these two notions seems to enjoy the…
Qixiao
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