Categories and functors, universal properties, algebras and algebraic theories, topoi, enriched and internal categories, structured categories (abelian, monoidal, etc), higher categories.

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4
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1answer
176 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
2answers
328 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
1answer
218 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
2answers
177 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
1answer
170 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
1answer
309 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
0answers
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
1answer
197 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
0answers
120 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
1answer
698 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
0answers
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
1answer
437 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
1answer
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
2answers
223 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
3answers
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
0answers
74 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
1answer
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
1answer
136 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
1answer
204 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
1answer
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
1answer
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
0answers
329 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
1answer
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
0answers
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?
5
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0answers
106 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
0answers
139 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 ...
1
vote
1answer
121 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
1answer
284 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
1answer
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
1answer
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 ...
4
votes
0answers
172 views

On the Axiom of Choice for Conglomerates and Skeletons

Say that $\mathcal{X}$ is a conglomerate if $\mathcal{X} = \{X_i: i \in I\}$, where each $X_i$ and $I$ are classes. The Axiom of Choice for Conglomerates is the statement: Whenever $\mathcal{X}$ and ...
2
votes
1answer
167 views

Set theoretic issue of localization of abelian categories

For a small abelian category in which every object is also a set, consider its localization with respect to a Serre subcategory (thus a quotient category), is it true that under this localization ...
14
votes
0answers
120 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 ...
21
votes
2answers
1k views

Lemma 1 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

Consider Lemma 1 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows. Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, $F: ...
7
votes
1answer
275 views

What are some examples of total derived functors that can't be computed from a functorial replacement?

(Or more generally, what are some examples of Kan extensions which are not pointwise?) By a total derived functor of a functor $F: C \to D$, I simply mean a (left or right) Kan extension of $F$ along ...
8
votes
0answers
129 views

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 ...
2
votes
0answers
118 views

For f: X -> Y -> X, what is the name of the property whereby for all x in X and y1, y2 in Y, f(f x y1)y2 = f(f x y2)y1?

My PL group has been discussing this (so we are really professional reject mathematicians, but this is more about the math behind what we are doing). Some of us called it a "generalized ...
1
vote
1answer
179 views

About the canonical morphism from $f^{*}f_{*}f^{*}F$ to $f^{*}F$

Assume $X$ and $Y$ are noetherian schemes over $\mathbb{C}$ and there is a proper and faithfully flat morphism $f: X\rightarrow Y$. Assume the canonical morphism $F\xrightarrow{\sim} f_{*}f^{*}F$ is ...
9
votes
1answer
217 views

Enriching categories and equivalences

Let $\mathcal{C}$ and $\mathcal{D}$ be two equivalent categories. Furthermore, assume $\mathcal{C}$ is enriched over a monoidal category $(\mathcal{M}, \otimes)$. Can one use the equivalence to ...
7
votes
1answer
253 views

Conditions on the fusion data of symmetric fusion category

We know that every symmetric fusion category (SFC) gives rise to data $N^{ij}_k$ that describe the fusion of simple objects: $i\times j = N^{ij}_k k$, and the data $\theta_i =\pm 1$ that describe the ...
7
votes
1answer
440 views

Does pullback in the category of smooth manifolds always exists?

I am looking for an example where $f:Y\to X$ and $f':Y'\to X$, are both smooth maps of smooth manifolds, but the pullback does not exist. Remarks: 1) A pullback in a certain category is defined as ...
24
votes
3answers
531 views

What is the point of pointwise Kan extensions?

Recall that a Kan extension is called pointwise if it can be computed by the usual (co)limit formula, or equivalently if it is preserved by (co)representable functors. I have seen pointwise Kan ...
3
votes
0answers
90 views

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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vote
0answers
181 views

Family $(X_y,D_y)$ with trivial canonical bundles

Let $i:D\hookrightarrow X$ and $f : X \to Y$ be holomorphic mappings of complex manifolds such that $i$ is a closed embedding and $f$ as well as$ f \circ i$ are proper and smooth and $D$ is a divisor. ...
37
votes
5answers
4k views

Are dagger categories truly evil?

Recall that a dagger category is a category equipped with an involution $*:Hom(x,y)\to Hom(y,x)$ that satisfies $f^{**}=f$ and $f^* g^*=(gf)^*$. A prominent example of a dagger category is the ...
11
votes
1answer
253 views

Reference request: sheaves on the site of d-manifolds

I believe I know how to prove the following results. I also know to whom to cite fancy-shmancy results that have these as a very special case. My question is: what are the correct citations for ...
10
votes
2answers
274 views

Is this a functor on the category of $C^{*}$ algebras?

The category of $C^{*}$ algebras is denoted by $\mathcal{A}$. Is there a functor $\mathcal{F}$ on $\mathcal{A}$ which send each object $A\in \mathcal{A}$ to its center $Z(A)$. In the other words, ...
7
votes
1answer
320 views

How (if at all) does category theory deal with situations where the usual notion of isomorphism isn't right?

This question will potentially rub some people the wrong way; I can't do much about this, except state right here at the outset that this question is motivated by a genuine desire to understand, and ...
6
votes
2answers
329 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 ...
11
votes
1answer
296 views

Elementary consequences of commuting limits and colimits over groups

This is a crosspost of this MSE question. In this n-cat cafe post, it is proven that for finite groups $G,H$ of coprime order, $G$-colimits and $H$-limits commute. Later on the following theorem is ...