# Tagged Questions

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)

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### What are some examples of cohomology theories without a corresponding classifying space?

The general nonsense of cohomology theories is that each one "should" be presented by a classifying space, so that maps into this space give the cohomology (before passing to connected components). ...
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### Weighted limits in the $Cat$-category of categories

What is a weighted limit in the $Cat$-category of categories, functors and natural transformations? I can find the general definition of a weighted limit for enriched categories in Kelly's book or ...
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### coequalizers and cofiber in a quasicategory

$\require{AMScd}$ Let $C$ be a stable $\infty$-category and let $f,f':X \to Y$ be two morphisms. Why is the coequalizer (in C, not hC) of $f,f'$ equal to the cofiber of $f-f'$? (This is claimed in the ...
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### in a pointed quasicategory, why is $X \simeq cofib(X[-1] \to 0)$? Answer: it's not

$\require{AMscd}$ Let $C$ be an pointed $\infty$-category (by which I mean quasicategory) admitting cofibers. For an object $X \in C$, why is $X \simeq cofib(X[-1] \to 0)$? Here $X[-1]=\Omega X$ is ...
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### Functors which induce isomorphisms on isomorphism-sets

Is there a name for functors $F : \mathcal{C} \to \mathcal{D}$ with the property that for all $A,B \in \mathrm{Ob}(\mathcal{C})$ the map $F : \mathrm{Isom}(A,B) \to \mathrm{Isom}(F(A),F(B))$ is an ...
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### Examples of n+1D TQFT with 1 dimensional Hilbert spaces on n-torus and n-sphere but higher dimensional Hilbert space on other n-manifolds

Are there simple examples of $n+1$D TQFT that assign 1-dimensional Hilbert spaces to both n-torus and n-sphere but higher dimensional Hilbert spaces to some other n-manifolds? Here I am assuming that ...
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### Maximum n-category constructible with a 1-category

Let $\mathcal{C}$ be a $1$-category. One can show that the category of internal categories in $\mathcal{C}$, with internal functors and internal natural transformations produce a $2$-category. I ...
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### How is the delooping of a groupoid constructed explicitly?

Let $G$ be a group, nlab's "delooping" page says that $G$ can be considered as a discrete groupoid in the $(\infty,1)$-topos $\infty$Grpd of $\infty$-grupoids, the delooping of $BG$ is then the ...
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### Name of this 2-categorical structure?

Let $\mathscr C$ be a 2-category. For each object $X$ in $\mathscr C$ suppose there is a category $\mathcal R(X)$, and for each pair of objects $X,Y$ in $\mathscr C$ there is an "action" functor  \...
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### Exponential objects of internal objects respecting evaluation (2-exponentials?)

Let $(F,+_F)$, and $(G,+_G)$ be two commutative internal monoids in Sets. Set being cartesian closed, I can form $G^F$ as a set. My question is simple: is there a canonical/universal way to enforce ...
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### Why are morphisms of monads lax and not oplax natural transformations

Monads in a bicategory $\mathscr B$ correspond to lax functors $* → \mathscr B$, so one expects morphisms of monads should correspond to nautral transformations between them. A natural ...
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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 ...
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### Homotopy groups of mapping spaces

If I have an $\infty$-category $\mathcal{C}$ (AKA quasi-category), can I say anything about the homotopy groups of the mapping spaces $\mathrm{Hom}_\mathcal{C}(X,Y)$ for two objects $X$ and $Y$? (...