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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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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How to define a weighted cone?

Let $F : I \to C$ be a diagram in $C$, and $N$ an object of $C$. A cone from $N$ to $F$ is a family of morphism $P_X : N \to F(X)$ such that for every morphism $f : i1 \to i2$ in $I$, $F(f) \circ P_X ...
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Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
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89 views

Construction of 2-limits in 2-categories

Limits in a category can be built as a combination of the basic limits that are products and equalizers. Is there a similar construction for 2-limits in a 2-category? If yes, is it from products and ...
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how do contravariant 2-functors preserve adjunctions?

I know that covariant 2-functors preserve adjunctions. Do contravariant 2-functors preserve the order left-right of the adjoints, or do they reverse it?
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modularizing category theory

I have made the experience that proofs using category theory often look very elegant and short but when it comes down to verifying the details there is quite a list of commutativities etc. to check. A ...
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Size issues in 2-categories

I was playing a bit the 2-category Cat trying to have a better understanding of the notion of a 2-category (strict I guess). The usual definition of a category that I use assumes that $Hom(A,B)$ is a ...
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Why is the category of n-categries cocomplete?

I am studying a variant of the notion of (strict) n-categories, and I would like to show that this makes a cocomplete category. For that I planned on adapting the proof of the cocompleteness of ...
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28 views

When is a functor of bicategories part of an equivalence?

Assuming the axiom of choice (I think), $F: \mathcal{C} \to \mathcal{D}$ is part of an equivalence of categories iff $F$ is fully faithful and essentially surjective on objects. This is sometimes ...
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What, if anything, does higher category theory have to say about situations where each subcategory induces a sub-$2$-category?

Let $\mathbf{Top}$ denote the $2$-category of topological spaces, continuous mappings, and homotopies between them. Let $\mathbf{C}$ denote a wide subcategory of $\mathbf{Top}$. Then we get a wide ...
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29 views

Quillen equivalence vs $\infty$-categorical equivalence

It's well known that a simplicial model category presents an $\infty$-category by the homotopy coherent nerve construction. (I am drawing my knowledge and terminology from what little of Lurie's ...
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128 views

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 ...
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A categorical perspective on the equivalence of sheaf cohomology and Cech cohomology?

In the nLab article on cohomology, I found the following passage. One can then understand various "cohomology theories" as nothing but tools for computing $\pi_0 \mathbf{H}(X,A)$ using the known ...
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Motivation for the nLab's definition of cohomology?

I am trying to penetrate the nLab article on cohomology. I don't know anything about higher category theory, but it seems like the real content here is topological. My question has two parts. First, ...
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73 views

Extended Topological Quantum Field Theory, (ETQFT) basics ..

What is the functorial (categorical) definition of a TQFT (Topological Quantum Field Theory), which Jacob Lurie "had extended", for his ETQFT ? Actually I just need to know what are basic tools, to ...
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Regarding the numbering of $n$-categories.

(By an $n$-category, I mean an $(n,n)$-category.) This is potentially a dumb question, but its been on my mind for a couple of years now, so I'm throwing in the towel and asking it. If it is a dumb ...
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41 views

Why are lax functors from the terminal $2$-category the same as monads?

I use boldface (e.g. $\mathbf{C},\mathbf{D}$) for categories and fraktur (e.g. $\mathfrak{B}$ for bicategories.) According to this blog post at nLab, with some of the notation changed a little: ...
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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 ...
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Smallest 3-category not equivalent to a strict 3-category

For modular lattices there is a canonical example of the "smallest" non-modular lattice, $\mathbf N_5$. Is there a similar example for 3-categories which are not equivalent to any strict 3-category? I ...
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Gerbes and Brauer group

Let $µ$ be a sheaf of abelian groups on a site $C$. There is a bijection between isomorphism (equivalence) classes of µ-gerbes over $C$ and $H^2 (C, µ)$. Can someone give me a good reference for the ...
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How to deal with adjoint pairs?

This may be a stupid question, but I just can't find out the solution. I'm confusion on how to deal with adjoint pairs, which means a pair of $1$-morphisms $(L,R)$ with two $2$-morphisms $\eta\colon ...
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56 views

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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definition of the notion of a subcategory of an $\infty$-category

Is the following formulation of the notion of a subcategory of an $\infty$-category ( - in the sense of section 1.2.11 of Lurie's Higher Topos Theory - ) correct?: Remark. Let $\mathcal{C}$ be an ...
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94 views

Intuition for the Dold-Kan correspondence

maybe this question does not make sense and it's just a psychological problem of mine. However I cannot understand the geometric picture of the Dold-Kan correspondence. Let $\mathbf{Ab}$ denotes the ...
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Infinitely “deep” globular sets / categories

In the usual sense a globular set consists of objects, arrows between objects, 2-arrows between arrows, etc. having $n$-arrows for every $n \in \mathbb N$ (although some may be empty). A category (or ...
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39 views

over categories of a morphism?

Given a category $\mathcal{C}$ and an object $x\in \mathcal{C}$ we can look at the over category $\mathcal{C}_{/x}$ whose objects are morphisms $d \rightarrow x$ for $d\in \mathcal{C}$ and morphisms ...
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definition of finite pointed spaces in Lurie Higher Algebra

Let $S$ be the infinity category of spaces. In Higher Algebra 1.4.1.4 Lurie defines $S^{fin}$ as the smallest full subcategory of $S$ which contains the final object $*$ and is stable under finite ...
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Why are split coequalizers “contractible”?

In the book Toposes, Triples and Theories by Barr and Wells, the authors define a contractible coequalizer (elsewhere known as a split coequalizer) to be a commutative diagram: $A ...
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34 views

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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37 views

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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Universal property and colimit

The free group $F(S)$ is the group given by a set $S$ with the universal property: For every group $G$ and map $f: S \to G$ there is a unique homomorphism $\phi: F(S) \to G$, such that $\phi \circ i = ...
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Why $|N(P_{i, j})| \cong [0, 1]^n$ as stated in page 21 of HTT?

maybe this is an idiot question, however I could not solve this after thinking for a while. I added the tag about higher categories simply because of the nature of the question, however this is just a ...
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coherence of inverses in 2-groupoids

Given a 2-groupoid $G$, two objects $a,b$, and two 1-morphisms $f,g:a\rightarrow b$ and a 2-morphism $\alpha : f \rightarrow g$, is it the case that there always exists a 2-morphism $\beta : f^{-1} ...
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On two definitions of the nerve of a simplicial category

Let ${\mathcal C}$ be a simplicial category. Then there are the following two ways of constructing a simplicial set from ${\mathcal C}$: Form the simplicial nerve $\text{N}_\Delta({\mathcal C}) := ...
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What's so special about the grounded poset of cardinality $2$?

By a grounded poset, I mean a poset $P$ with a bottom element $0_P$. Arrows between grounded posets are monotone mappings that preserve the basepoint. This gives a self-enriched category; lets call it ...
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69 views

What is a Monad in the two category $Rel$?

The 2-category $Rel$ is a category with sets as $0$-cells, relations as $1$-cells (with relation composition as composition), and inclusions as $2$-cells (with vertical composition being the fact that ...
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53 views

Is Module Category over Monoidal category Monoidal?

let $\mathcal{C}$ be a monoidal category and $\mathcal{M}$ a $\mathcal{C}$-module category. Does $\mathcal{M}$ need to be a monidal category? I know it is true for certain categories, but is it true ...
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The bicategory Bibun

Are all the $2$-morphisms in the bicategrory (of Lie groupoids, right-principal bibundles and bibundle morphisms) Bibun isomorphisms?
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Relationship between differential cohesion and synthetic differential geometry

I was wondering what is the relationship between differential cohesion and synthetic differential geometry? I know the basics of synthetic differential geometry from Kock's text, but I am not ...
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150 views

Yoneda Lemma for 2-categories - lax version

Is there a sort of lax Yoneda Lemma for 2-categories? Here is what I seem to have proven (although I have not checked all the details): If $\mathcal{C}$ is a (weak) 2-category, $A$ is an object of ...
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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 $Y$? ...
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Should an object in the category always be a formal mathematical structure?

Today I heard during a popular lecture about the applications of category theory, than an object in the category should always be some kind of mathematical structure e.g. a set, ring, monoid, etc. So ...
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List of universal properties

At the moment I am looking into category theory and I am wondering if there exists a list of universal properties? I couldn't seem to find one.