# Tagged Questions

29 views

### How does a section of a stack give a sheaf?

At nLab in the article constant stack and a few other related articles, a pattern is mentioned where a section of a constant sheaf is a locally constant function, a section of a constant stack is a ...
39 views

### Are there any stable $(\infty,1)$-topoi?

Can a stable $(\infty,1)$-category be an $(\infty,1)$-topos?
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### Homotopy direct limit versus direct limit

Let $X_1\to X_2\to \cdots$ be an infinite sequence of maps. Then, as I understand, the homotopy direct limit of this sequence is the space formed by taking the disjoint union of the $X_i\times I$ and ...
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### Does the lax Gray tensor product preserve fully faithful 2-functors?

The question is in the title. By fully faithful 2-functor, I mean 2-functors such that the maps on the hom categories are isomorphism, and by preserve, I mean in each variable. I have an argument ...
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### Stacks versus sheaves with values in categories

A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to ...
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### Defining a monoidal category without elements

I am trying to generalize the notion of monoid object internal to a (not necessarily strict) monoidal category, by weakening the associativity and unitarity diagrams (see this nlab entry.) Of course ...
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### n-globular sets and n-categories

Is the forgetfull functor between the category of n-categories and the category of n-globular sets always monadic? It's seems so, but, in nLab, they are talking only about "2-globular sets and ...
89 views

### 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 ...
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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 ...
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It is well known that a category is the same as a monad in the 2-category of spans. So I am wondering if there is a similar statement hold for higher categories: can a bicategory be given as a weak ...
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### Distributivity in linear monoidal categories

Let $\mathcal{C}$ be a linear monoidal category, that is a monoidal category (with tensor product $\otimes$) enriched over $\mathbf{Vect}$. Now as far as I can tell the axioms for a linear category ...
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### Derived pseudo-functor

Let $\mathfrak {X}\to \mathfrak{Y}$ be a pseudofunctor (in which $\mathfrak{X}$ is a model category and $\mathfrak{Y}$ is a bicategory). I would like to understand when there is a derived functor ...
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### What is the name and properties of a category C with hom(C) is subclass of ob(C)?

Suppose we have a category $C$ such as $hom(C) \subset ob(C)$ (so every arrow $f: a \rightarrow b$ of $C$ is also an object of $C$). What is the name and the main properties of such category?
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### Difference between the simplicial nerve and the nerve of a simplicial category

In Jacob Lurie's Higher Topos Theory book, he defines the following notion of a simplicial nerve: Definition 1.1.5.5. Let $\mathcal{C}$ be a simplicial category. The simplicial nerve ...
104 views

### $2$-Morphisms in the Fundamental $2$-Groupoid

I'm trying to write down a clean definition of the fundamental $2$-groupoid $\pi_{\leq 2}(X)$ of a topological space $X$. Specifically, I'm concerned with how to properly define $2$-morphisms. Here is ...
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### Totally ordered set Category

If 2 is the totally ordered set, and C is any category, this is given F is a functor from $2 \to C$ then what type of objects and arrows of the functor between them. As far as I understand as 2 is ...
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### Higher categorical Yoneda lemma

One of the most powerful results in ordinary category theory is the Yoneda lemma, and so the following question seems natural: Is there an analogue of the Yoneda lemma for (weak) $n$-categories? I ...
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### Can the equivalence between principle bundles and maps to classifying spaces be turned into an adjunction.

We have that $G-PBun(X)$, the category of topological principal bundles for a structure group $G$ is equivalent to $Top[X,BG]$ where $BG$ is the classifying space of $G$. This almost looks like an ...
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### Has any NNO been indentified in an infinity topos?

The Natural Numbers Object ($NNO$) in $Set$ is just the Integers. Have any been identified or conjectured in $(\infty,1)$-topos that isn't an ordinary topos, in particular it could be a $(2,1)$-topos. ...
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### How to understand the definition of sets in homotopy type theory and the role of univalence?

Bear with me, I'm a physicist. In homotopy type theory, as I understand it, a type $X$ is a set if all the morphisms over its terms $x:X$ are identies. When I say "morphisms", then I view the term as ...
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### What is homotopy in $(\infty,1)$-categories (as weak Kan complexes)

One of the peculiar (and somewhat appealing) features of quasi-categories is that many properties from ordinary category theory characterized equality are characterized by some form of homotopy ...
43 views

### Reference request for 2-category theory

In the theory of 2-categories, we have the following theorem. A pseudofunctor is an equivalence of 2-categories if it is essentially surjective on objects, and the induced functor on Hom categories ...
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### What does the following notation means in the context of 2-categories? $\bullet$

I know the elementary definition of an adjunction and the compact version (i.e. in terms of naturality of two hom-sets) but I am reading a definition which I cant understand its notation, namely the ...
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### How to see a 2-group as a 2-category with only one object?

We'll take the following definition of a 2-group: A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$ Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of ...
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### Two properties of the category $Pr^L$

This is a continuation of another question about the symmetric monoidal category of locally presentable categories; now for the (โ,1)-analogue. Googling a bit I found two astonishing (albeit quite ...
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### From where comes the horizonal composition in a 2-category?

Viewing a 2-category as a category enriched in $\mathsf{Cat}$, I can see from where comes the vertical composition: morphisms of a 2-category are objects of $\mathsf{Cat}$ and 2-morphisms of this ...
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### Is it true in a presentable infinity category that algebras are homotopy colimits of free algebras?

In 1-categorical algebra one knows that, in a locally presentable category, every algebra for a finite product theory is a colimit of free algebras. Is the same true for algebras of finite product ...
166 views

### Is it possible to formalize a “universe” of categories as a one-sorted first-order theory with one binary relation and no functions?

This is a modification of a question I asked earlier. In that question, I hadn't placed any limits on the number of binary relations allowed, so my question had an affirmative answer, but a trivial ...
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### What is an object classifier and how does give a natural numbers object?

According to a history of topos theory by McLarty, Blass (1989) showed that the existence of an object classifier over a given topos implies that the topos has a natural number object. What is an ...
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### 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 ...
236 views

### What was the Lawveres explanation of adjoint functors in terms of Hegelian Philosophy?

I was contemplating asking this question on Philsophy.SE but felt it was better directed here as there are a dearth of category theorists there. According to the wikipedia entry on Categorical Logic: ...
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### What is a (-1)-morphism?

So, I read the John Baez essay "Lectures on n-categories and cohomology" and I understand the notion of a (-1)-category" and a (-2)-category" and how to derive them. However, I'm not totally clear on ...
134 views

### What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
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### Are lax kernel pairs stable under change of base?

The question is immediate for kernel pairs, but when we work in a category enriched in posets, like the category of locales is, are the lax kernel pairs stable under change of base, or since perhaps ...
207 views

### What is a monad in a $2$-category?

The wikipedia article on monads somewhat mysteriously notes that Monads can be defined in any 2-category ${\mathfrak C}$. The monads defined above are for ${\mathfrak C}$ = Cat. where Cat is the ...
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### Are bicategories and lax 2-categories the same?

My question is that whether the definition of bicategories is the same as the definition of lax 2-categories. I heard that they are both week versions of 2-categories. Are they the same? If not, how ...
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### 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 ...
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### Deligne tensor product

I would like to know something about a tensor product of categories and it seems Deligne tensor product is what I am looking for. But the paper "Categories tannakiennes" by Deligne is not available ...
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### 2-morphisms from spans of spans

I have a question about the construction of 2-morphisms from spans of spans in the paper "2-vector spaces and groupoid" by Jeffrey Morton . Suppose we have a span of span of groupoids as follows and ...
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### Basics of Lie 2-algebras?

Could somebody (simply) explain the basics foundations of Lie 2-algebras, and some basic practical applications ? For instance, does it exist a 3-map (equivalent to the 2-map commutator for Lie ...
106 views

### 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, ยง7.2] comprises three objects $C_0, C_1, C_2$ in $\mathcal{S}$ and ...
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### Dual of a 2-category

Let $C$ be a $2$-category. There are two ways of dualizing $C$: The first one is well-known and also generalizes to arbitrary $(\infty,1)$-categories: We dualize "at each stage". The second one only ...
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### Do the terms “quiver” and “meta graph” refer to the same concept?

Do the terms "quiver" and "metagraph" refer to the same concept? Or is there a distinction I am missing. My sources are Quiver - http://ncatlab.org/nlab/show/quiver Metagraph - ...
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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} ...
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### Morphisms with an arbitrary number of objects

Is this structure familiar for you? It consists of a category $C$ a set $M$ a function $\operatorname{arity}$'' defined on $M$ a function $\operatorname{Obj}_m$ defined for every ...
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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 ...
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### How to define a 2-category

I found this definition in http://arxiv.org/abs/hep-th/0304074 (pp.13) for 2-categories: It is a collection $\mathcal{C}_0$ of objects, $\mathcal{C}_1$ of morphisms, and $\mathcal{C}_2$ of 2-morphisms ...
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### Automorphisms and bicategories

I'm reading Lang's section on field theory and he stresses that, unlike typical "universal" constructions which are determined up to unique isomorphism, algebraic closures (and by extension, their ...
### $(\mbox{Sh,Sh-map})$ represents the category of sheaves on a stack
I'm trying to understand the following theorem, and I think I don't understand the definitions. Let $(\mathcal{C},J)$ be a site (with a subcanonical topology). Write $\mathcal{C}/X$ for the groupoid ...