The higher-category-theory tag has no usage guidance, but it has a tag wiki.

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### Why is every object cofibrant in an excellent model category?

In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...

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152 views

### Interaction of Grothendieck Construction with Coherent Nerve

There are a number of Grothendieck constructions: one for discrete categories, one for enriched categories (see Tamaki's paper here) and one for quasicategories (see the Unstraightening and ...

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**1**answer

463 views

### Errata on Rezk's paper

I am reading this paper of Rezk's http://arxiv.org/abs/0901.3602 A cartesian presentation of weak n-categories, and as it is pointed out in the introduction, it contained a wrong statement (2.19 in ...

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votes

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131 views

### Set of functions is not a bifunctor on Rel

Let Rel be the category whose objects are sets and whose morphisms are binary relations, with composition defined by $x (S \circ R) z \Leftrightarrow (\exists y : x R y \wedge y S z)$, and identity ...

**10**

votes

**1**answer

163 views

### Which simplicial objects are Čech nerves?

In 1-categories, a regular epimorphism is a coequalizer of some parallel pair. An effective epimorphism is one which coequalizes its kernel pair. In the presence of kernel pairs, regular and ...

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votes

**1**answer

136 views

### Appropriate morphisms and 2-morphisms in Ind(C)

As I was trying to understand the category $Ind(C)$ of diagrams of the form $I \to C$, where $I$ is a small filtered $(0,1)$-category, I wondered whether it is possible to define morphisms directly, ...

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**3**answers

480 views

### “Spatial (geometrical)” realization of Elementary topos?

It is well known that (Grothendieck) Topos (in fact, Model topos too) has many good geometrical properties. In many senses reflects general forms of generic geometry.
Note: Grothendieck view of Topos ...

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votes

**1**answer

91 views

### Can a weak fibration category be non saturated?

A weak fibration category is a category $\mathcal{C}$ equipped with two subcategories
$$\mathcal{F}, \mathcal{W} \subseteq \mathcal{C}$$
containing all the isomorphisms, such that the following ...

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**1**answer

291 views

### Difficulties with descent data as homotopy limit of image of Čech nerve

Apologies if this question is inappropriate for MO. It is not a research level question in any of the topics it addresses, I just don't see how a novice can go about answering it alone (I've tried ...

**5**

votes

**1**answer

189 views

### What do globes (used to construct globular sets, $\omega$-categories, etc.) actually look like?

Nlab introduces the globular category as a geometrical model to construct certain higher categorical structures (e. g. strict $\omega$-categories), just as quasi-categories, for example, are modelled ...

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316 views

### Coherence and rewriting

In category theory there are numerous coherence theorems (https://ncatlab.org/nlab/show/coherence+theorem). One example is the Mac Lane's coherence theorem for monoidal categories. This and probably ...

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**1**answer

233 views

### Relative version of Quillen's theorem A

Quillen's Theorem A is formulated as follows:
Let $F:X\to Y$ be a functor between small categories. Suppose for each $y\in Y$ the category
$F/y$ is contractible. Then $F$ induces a weak equivalence ...

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vote

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45 views

### Pseudopullback of dimension three

What is the name of the appropriate analogue of the pseudopullback for dimension three?
That is to say, a pseudonatural equivalence $fg\simeq hj $ which is universal in the obvious sense...
Thank ...

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votes

**2**answers

2k views

### What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?

In 1991, Kapranov and Voevodsky published a proof of a now famously false result, roughly saying that the homotopy category of spaces is equivalent to the homotopy category of strict infinity ...

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votes

**1**answer

58 views

### Why does vertical multiplication in 2-groups not follow the same order as horizontal one when constructed from crossed modules?

Consider a 2-group (seen as a 2-category with only one object $\star$) constructed from a crossed module $(G,H,t,\rhd)$ ($\circ$ will denote 1-morphisms composition, $\circ_{h}$ 2-morphisms horizontal ...

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92 views

### Stability of adjunctions of infinity-categories by base change

Let $O \to O'$ be a functor between locally presentable symmetric monoidal $(\infty,1)$-categories (assume that the tensor product commutes in each argument with colimits, if necessary). Suppose that ...

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**1**answer

667 views

### How to stop worrying about enriched categories?

Recently I realized that ordinary category theory is not a suitable language for a big portion of the math I'm having a hard time with these days. One thing in common to all my examples is that they ...

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91 views

### What terminology surrounds “involutive” double categories?

Write $\mathbf{Cat}$ for the world of categories. Then $\mathbf{Cat}$ has:
objects (namely cateories)
arrows (namely, functors)
proarrows (namely, bimodules)
squares (namely, functors between pairs ...

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79 views

### What do you call the coherence cells in a lax morphism?

The original question a friend asked me is what to call the coherence cells in a lax monoidal functor. After looking around, I was surprised to realize that when it comes to monoidal functors, ...

**5**

votes

**1**answer

123 views

### Uniqueness of $\infty$-adjoints

Adjoints in a 2-category are essentially unique, in the following strong sense. If $\mathbf{2}$ denotes the "walking arrow" category $(\cdot \to \cdot)$, then there is a 2-category $\mathrm{Adj}_1$ ...

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votes

**1**answer

176 views

### Switching left and right adjoints in recollement situations

In Fascieaux pervers, Beilinson, Bernstein and Deligne define a recollement situation as a triple of triangulated categories $\mathcal{D}_U,\mathcal{D}_F$ and $\mathcal{D}$, together with functors
$$
...

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235 views

### Is the hom-simplicial set in the hammock localization a nerve?

Let $(C,w)$ be a relative category. Then associated to it we have its hammock localization, $L^H(C,w)$, which is a simplicially enriched category.
If $X,Y\in C$, the description of the simplicial set ...

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votes

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176 views

### The source-side-opposite of the arrow category

Given a category $C$, is there a name for the following category:
$\mathrm{Obj}(D) = \left\{ (x, y, f) \middle| x, y \in \mathrm{Obj}(C), f \in C(x, y) \right\}$
$D((x, y, f), (x', y', f')) = \left\{...

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45 views

### Holonomy 2-functor transformation by transition functions

The holonomy 2-functor on a $\mathcal{G}$-principal 2-bundle associates a bigon:
$$\mathsf{hol}_i(\Sigma):\mathsf{hol}_i(\gamma)\Rightarrow \mathsf{hol}_i(\gamma')$$
in $\mathcal{G}$ to each bigon:
$$\...

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178 views

### Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?

Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined
$$
RHom(C,...

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150 views

### Action of a strict 2-group on a category gives autoequivalences?

A strict action of a strict 2-group (seen as 2-category with only one object $\star$) $\mathcal{G}$ on a 2-space (principally a category) $\mathcal{M}$ is a strict 2-functor $\Phi:\mathcal{G}\to \...

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244 views

### strict 2-groups VS crossed modules

nLab defines a strict 2-groups in many different but equivalent ways, among them:
an internal group object in Cat,
an internal group object in Grpd
Also, it is known that strict 2-groups may be ...

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votes

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273 views

### derived categories as presentable DG-categories

Let $A$ be a ring. Is it true that the DG category of unbounded complexes of $A$-modules, localized by quasi-isomorphisms, is cocomplete and compactly generated? What would be a reference for that and ...

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193 views

### Does every bicategory have a “delaxing object”?

If I'm not mistaken, there is a bicategory $\mathsf{Monad}$ given as follows:
Start with the associative operad.
Deloop it to obtain a multicategory.
Adjoin objects and morphisms as necessary to ...

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192 views

### A definition of the homotopy colimit of a coherent diagram

Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in ...

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**1**answer

94 views

### A question on 2-bundles

In this paper, the authors +John Baez and +Urs Schreiber defined (page 15) "transition functions" for a special kind of 2-bundles (those whose the base space is a ordinary smooth space augmented to a ...

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317 views

### When does algebraic K theory behave like a cohomology theory

Let $\mathbb{F}$ be a field. Let $K(\mathbb{F})$ be its algebraic (Quillen) $K$-theory spectrum. Let $X$ be a (nice, finite CW) topological space and let $\text{Rep}\Omega(X)$ be the DG category of (...

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107 views

### Category enriched over a monoidal 2-category

Consider a monoidal 2-category (or bicategory) B. For example, B could by the 2-category (finite sets, finite correspondences, isomorphisms of correspondences) with monoidal structure given by ...

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188 views

### Operadic Lift of Lurie's Relative Tensor Product

In Section 4.4 of his book Higher Algebra, Lurie introduces, for a monoid object $A$ of a monoidal quasicategory $C$, and right and left $A$-modules $M,N$, the relative tensor product $M\otimes_AN$. ...

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252 views

### Model independent proof of colimit formula for left Kan extensions

I am interested in finding a proof of the colimit formula for left Kan extensions $(\infty,1)$-categories which does not rely on a chosen model. (By a model inpendent proof I mean a proof which uses ...

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178 views

### Equivalence of natural transformations

Let $\mathcal{C}$ be a small category and $\mathrm{Cat}$ be the 2-category of small categories.
Let $F,G : \mathcal{C} \to \mathrm{Cat}$ be two functors and $\theta : F \to G$ be a natural ...

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votes

**1**answer

173 views

### Definition of Left Operadic Kan Extension for $\infty$-operads

In Lurie's book Higher Algebra, he makes the following definition:
Definition 3.1.2.2: Let $M^\otimes\to N(Fin_\ast)\times\Delta^1$ be a correspondence from an $\infty$-operad $A^\otimes$ to another $...

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70 views

### How to show these class of morphisms are perfect

In Lurie's book "Higher Topos Theory", a class $\mathsf{W}$ of morphisms in a category $\mathcal{A}$ is called perfect if
Every isomorphism belongs to $\mathsf{W}$.
it satisfies "2 out of 3 ...

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votes

**1**answer

442 views

### Automorphisms of Eilenberg-Mac Lane spaces and semidirect products (and the odd line)

If $A$ is an abelian group, we have
$Aut\left(K\left(A,n\right)\right)=Aut(A) \ltimes K\left(A,n\right),$
where the left hand side is the space of self-homotopy equivalences. Is there an easy way to ...

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votes

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223 views

### Geometric morphism of $\infty$ topos

I have a very simple question regarding geometric morphisms of $\infty$ topoi, but have been unable to find the answer in Lurie's HTT (although it seems likely that its there somewhere and I just can'...

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### How to realize the descent data of Qcoh as a (pseudo)-limit in Cat?

It is well-know that $Qcoh$ is a fibered category on $Sch$. In more details let $\mathcal{C}$ be the category $(Sch/S)$ of schemes over a fixed base scheme S. For each scheme $U$ we define $Qcoh(U)$ ...

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128 views

### Join of simplicial categories

Let $\mathcal{C},\mathcal{D}$ be simplicial categories.
Of course, we have the "naïve" join $\mathcal{C} \star \mathcal{D}$, which has
$$
\mathrm{Ob}(\mathcal{C} \star \mathcal{D}) := \mathrm{Ob}(\...

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128 views

### Specific cases of the tangle hypothesis in terms of “classical” n-categories

As is well known, the tangle hypothesis of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal $n$-...

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### A completeness criterion for $\infty$-categories

We all know that for ordinary categories $\mathscr{C}, \mathscr{D}$ (with $\mathscr{C}$ small) the limit of a functor $F:\mathscr{C} \to \mathscr{D}$, if it exists, can be constructed by using ...

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133 views

### Differentiation of Lie $\infty$-groupoids

I've been trying to understand how to differentiate Lie $\infty$-groupoids to get a Lie $\infty$-algebroid. First of all, I will state the definitions that I'm assuming.
A Lie $\infty$-groupoid is a ...

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### Colimits of n-fold categories

An $n$-fold category is an internal category in the category of $(n-1)$-fold categories (and a $0$-fold category is just a Set).
General results about internal categories assure that the category of $...

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341 views

### Is dgCat a category or a 2-category?

Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e. dgCat has ...

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### About the definition of lax.functor between tricategories

SUMMARY: Observing that monoids in a monoidal category are identified with lax.functors (with domain 1), I tried to generalize this argument wanting to get a skew-Monoidal-category as (tri)lax....

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### Minimal model (resolution) for a specific colored operad

We know that for the operad $As:=\mathcal{F}(\mu)/(\mu\circ_1\mu-\mu\circ_2\mu)$, its minimal model is the free operad $\mathcal{F}(E)$ where $E=\mathbb{k}<\mu_2,\mu_3,\dots,\mu_n,\dots>$ is the ...

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### Cancellation property of groupoidal cartesian fibrations

I have an issue concerning a property of "left cancellation" for groupoidal cartesian fibrations of ∞-cosmoi (but everything works fine in a 2-category as well).
A 1-cell $p: E \to B$ is called ...