**3**

votes

**1**answer

106 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 ...

**22**

votes

**2**answers

1k views

### “Why” are monadicity and descent related?

This question is probably too vague for experts, but I really don't know how to avoid it.
I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the ...

**8**

votes

**0**answers

140 views

### Which nice subcategories of $\mathsf{Top}$ are locally cartesian closed?

For a class $\mathcal{J}$ of topological spaces, let $\mathsf{Top}_\mathcal{J}$ denote the category of $\mathcal{J}$-generated spaces, i.e. those spaces $X$ such that $U\subseteq X$ is open iff ...

**16**

votes

**2**answers

436 views

### What categorical property of monoidal categories picks out the ones with duals?

Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...

**21**

votes

**1**answer

824 views

### Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I

In his paper Higher Algebraic K-Theory I (see [here][1]), Quillen introduces a topological space $BC$, called the classifying space of $C$, and tries to relate its topology to the categorical ...

**2**

votes

**1**answer

161 views

### Faithful exact functors to tensor categories

Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P $ an additive $k$-linear exact and faithful functor. I want to ...

**6**

votes

**0**answers

224 views

### Many-sorted nominal sets as sheaves

The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...

**15**

votes

**1**answer

590 views

### Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction),
as defined by Grayson in Higher algebraic K-theory: II (page 219),
takes as an ...

**3**

votes

**0**answers

190 views

### Defining Inertia Stack

Let $X$ be a topos and $F: \zeta \rightarrow X$ a stack on $X$. Now in the paper http://arxiv.org/pdf/math/0411337v2.pdf ( Definitions 2.1.1.5 and 2.1.1.1) of Lieblich, he describes the inertia stack ...

**6**

votes

**1**answer

251 views

### What is known about the large cardinal strength of Shelah's categoricity conjecture?

Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of ...

**6**

votes

**1**answer

364 views

### P.G.Goerss, J.F.Jardine, “Simplicial Homotopy Theory” prerequisites

I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow.
Besides, I know that there ...

**4**

votes

**0**answers

113 views

### Topos Theory, internal Heyting Algebra

Given a topos $\mathcal{E}$ with subobject classifier $\Omega$.
If we denote by $N\Omega$ the former of all local operators on $\Omega$, that is, Lawvere–Tierney topologies of $\mathcal{E}$, it is ...

**3**

votes

**1**answer

122 views

### What is an example of a colimit-dense generator which is not dense?

An object $G$ of a category $\mathcal{C}$ is a dense generator if every object $X$ is the colimit of the canonical diagram of copies of $G$ mapping to $X$. (This canonical diagram is indexed by the ...

**3**

votes

**1**answer

167 views

### Left adjoint of pullback

In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states:
Indeed, the UMP of pullbacks essentially states that composition along
any function α is left adjoint to pullback ...

**10**

votes

**1**answer

514 views

### Categorical or simplicial introduction to modern homotopy theory

I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering...
...

**9**

votes

**0**answers

198 views

### $\mathcal{M}(\mathcal{D}_X)$ and $\mathcal{M}^r(\mathcal{D}_X)$ have natural tensor category structures?

Write $\mathcal{M}^\ell(\mathcal{D}_{X/S}) = \mathcal{M}(\mathcal{D}_{X/S})$ for the category of left $\mathcal{D}$-modules over $X$ and $\mathcal{M}^r(\mathcal{D}_{X/S})$ for the category of right ...

**1**

vote

**0**answers

141 views

### Coinduction in a presheaf topos?

Fix a presheaf topos $\mathcal{E}\triangleq \mathbf{Set}^{\mathcal{C}^\mathsf{op}}$. Then, for any object/presheaf $X:\mathcal{E}$, I can define the presheaf of relations on $X$ as ...

**2**

votes

**2**answers

301 views

### In a fibration, where does the generic object live?

In Bart Jacobs. Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, 1999. isbn: 9780444508539, the author writes, p. 326:
Sometimes, for ...

**3**

votes

**0**answers

183 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 $M\otimes_AN$. ...

**6**

votes

**0**answers

97 views

### Finitely presented algebras with isomorphic semilattices of congruences

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward ...

**4**

votes

**1**answer

177 views

### Exterior derivative as only (up to multiple) natural operator $\Lambda ^kT^\ast \rightsquigarrow \Lambda ^{k+1}T^\ast$

In Kolar, Michor, & Slovak's book Natural Operations in Differential Geometry, it is proved the exterior derivative is universal in the following sense.
Proposition 25.4. For $k>0$ all natural ...

**4**

votes

**2**answers

331 views

### Natural operators in differential geometry - why are they natural?

I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of ...

**5**

votes

**1**answer

225 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 ...

**4**

votes

**2**answers

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 ...

**3**

votes

**1**answer

171 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 ...

**14**

votes

**1**answer

315 views

### Does the category of categories-mod-natural-isomorphism have any nonobvious autoequivalences?

The simpler question is to study the 2-groupoid $\mathrm{Aut}(\mathsf{Cat})$ of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories ...

**3**

votes

**0**answers

62 views

### Bi-adjointness and isomorphisms of (co)limits

Without getting into too many (presumably irrelevant) details, I have a functor $F:A \to B$ between reasonably mellow categories so that for each object $b \in B_0$ the universal morphism
$$g_b: ...

**4**

votes

**1**answer

198 views

### What is the right notion of generalized element of a category?

I've been working out how the internal language of a category C extends to taking the category itself as a type.
The most obvious way to interpret $X : \mathbf{C}$ is, of course, that $X$ is an ...

**3**

votes

**0**answers

121 views

### Topologized category of bounded chain complex

I am reading Segal's paper 'categories and cohomology theories' [1], but there is one claim (in the last example in sec.2) I don't quite understand:
Let $\mathcal{C}$ be the category of bounded chain ...

**20**

votes

**1**answer

708 views

### Lemma 2 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

This is a followup to here.
Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows.
Lemma 2. For any pair $i$, $j$ such that $0 ...

**2**

votes

**0**answers

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 ...

**12**

votes

**1**answer

440 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 ...

**5**

votes

**1**answer

260 views

### Not sure how to fix an error in the Handbook of Categorical Algebra (vol 3)

In the proof of Theorem 1.5.7 (in which it is shown the the nuclei on a locale, when ordered by pointwise partial order, themselves form a locale) in the computation at the bottom of p.35, there is ...

**5**

votes

**2**answers

227 views

### global section of affine $C^\infty$-scheme

I'm reading Algebraic Geometry over $C^\infty$-rings.
It is written that "If $\mathfrak{C}$ is not finitely generated then $\Phi_{\mathfrak{C}}:\mathfrak{C}\rightarrow ...

**41**

votes

**3**answers

1k views

### Duality between Compactness and Hausdorffness

Consider a non-empty set $X$ and its complete lattice of topologies
(see also this thread).
The discrete topology is Hausdorff. Every topology that is finer than a Hausdorff topology is also ...

**2**

votes

**0**answers

75 views

### On subdirect products [closed]

I'm sorry if this question doesn't fit with MO rules, but I've asked on Math SE yet without answers, so I post here with the hope someone will answer me.
I want to show, knowing the Goursat's ...

**6**

votes

**1**answer

152 views

### internalization of the concept of large and small category

I have been poking around the internet and nlab looking at the concept of large and small categories. My original focus was locally presentable categories of categories and I was thinking of finding ...

**3**

votes

**1**answer

139 views

### Twists, balances, and ribbons in pivotal braided tensor categories

Let $\mathcal{C}$ be a pivotal tensor category. Feel free to assume finiteness, semisimplicity, fusion, sphericality, unitarity or whatever makes things interesting. Which of the following structures ...

**8**

votes

**1**answer

209 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 ...

**5**

votes

**1**answer

124 views

### Monoidal cats and string diagrams for a semantics of object oriented programming languages

It seems to me that in a statically typed, object oriented language, there is a striking similarity to wiring diagrams. Wires (objects) of type $X$ go into functions (boxes) of input type $X$. Is ...

**3**

votes

**1**answer

188 views

### Reference for constructing tensor products of finitely presented functors

I need references related to the construction of tensor product between functors
Let $k$ be a commutative ring, $C$ a small $k$-linear category and $A$ cocomplete abelian category. Let $A^C$ denote ...

**12**

votes

**0**answers

330 views

### Is there a Galois theory for $\mathbb R_{\geq 0}$?

The broadest version of my question is the following:
Where can I find algebrogeometric abstract nonsense that handles "rings" and "fields" like $\mathbb R_{\geq 0}$ in which there is no ...

**2**

votes

**1**answer

154 views

### Formal DG-algebra

Let $\mathcal{C}$ be a nice $k$-linear abelian category (the example I have in mind is the category of coherent sheaves on a smooth projective variety over $\mathbb{C}$). Let $B \in ...

**5**

votes

**0**answers

69 views

### Uniqueness of lax 2-adjoints

In this nlab page there is defined the notion of a lax 2-adjoint. My question is to what extent would an adjoint of this type be unique?

**6**

votes

**0**answers

108 views

### 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 $Qcoh(U)$ ...

**6**

votes

**0**answers

143 views

### When are the categories of algebras over props (co)complete?

Suppose P is a (colored) prop in a closed symmetric monoidal locally presentable category C. Is the category Alg_P of algebras over P in C locally presentable?
It seems that one can relatively easy ...

**2**

votes

**1**answer

123 views

### The definition of unitary fusion category

I just come across a definition of the unitary fusion category:
A fusion category $\mathcal{C}$ over the complex number is said to be unitary if we have:
We have a Hilbert space structure on each ...

**-1**

votes

**1**answer

288 views

### Representable functors and direct limits

Let $\mathcal{F}:\mathrm{Sch}/S \to \{\mathrm{Sets}\}$ be a representable functor. Denote by $X$ the scheme representing $\mathcal{F}$. The question is whether the natural tranformation ...

**0**

votes

**1**answer

132 views

### Creating Duals in A Category

Before stating my question I would like to provide afew motivating examples:
Examples:
In the category of Finitely-generated-projective $R$-modules, we have that:
$M^{\vee}:=Hom_R(M,R)$ satisfies: ...

**0**

votes

**1**answer

71 views

### A confusion about covering flatness

I'm reading this entry on nLab. But I'm confusing with the notion of covering-flatness. More precisely, I meet some trouble when I try to show that the $Sets$-valued flatness is a special case of ...