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

4
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0answers
41 views

Proposition in HTT on cofibrations of categories

Proposition A.3.3.9. in Higher Topos Theory is as follows: Let $S$ be an excellent model category and let $f:C\rightarrow C'$ be a cofibration of small $S$-enriched categories. Then (1) for every ...
0
votes
0answers
65 views

Does the 2 category of Groupoids Admit the Vector Space Monad?

We can see here in Jacob's 2013 paper, that he seems to state that a particular kind of multiset monad is actually a vector space monad. 3.2. Vector spaces. For a semiring S one can define the ...
4
votes
1answer
102 views

What is the idempotent completion of the (2,1)-category of spans of finite sets?

I don't believe the $(2,1)$-category $FinSpan$ has split idempotents. Question: Is there a simple description of the idempotent completion of $FinSpan$? Foundationally, we may think of $FinSpan$ as ...
5
votes
0answers
65 views

Internal hom as 2-Kan lift of pseudofunctor

Consider a situation where there is a pseudofunctor from some category $C$. Due to it's geometrical properties, it might induce some sort of an internal hom on the original category (making it a ...
2
votes
2answers
81 views

Example: Accessible category without colimits

I am looking for intuitive examples of the way(s) that colimits may fail to exist in the category of (Set-valued) models for a limit/colimit sketch. Bonus points if the sketch and/or the colimit ...
7
votes
2answers
91 views

What is a true invariant of $G$-crossed braided fusion categories?

Definition. An invariant of a (spherical) fusion category with extra structure is a number or a set or tuple of numbers preserved under (appropriate) equivalences. (Spherical) fusion categories have ...
4
votes
0answers
133 views

$MK+CC$ as a foundation for category theory

Has any work been done on what $MK+CC$ looks like as a foundation for category theory? Is it 'the same' as restricting to inaccessibles in some precise manner? According to wikipedia, any category ...
2
votes
0answers
120 views

Is Quillen's bracket a “universal enveloping” something?

$\newcommand{\G}{\mathcal{G}}$ In K-theory, there is a construction due to Quillen as follows. Let $(\G, \oplus, 0)$ be a monoidal groupoid. Then the bracket $\langle \G, \G \rangle$, sometimes also ...
6
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0answers
102 views

Product-preserving fibrant replacement functor for the Joyal model structure

There are a few fibrant replacement functors for the Quillen model structure on simplicial sets that preserve finite cartesian products, namely $\operatorname{Ex}^\infty$ and $\operatorname{Sing}(|\...
-2
votes
2answers
210 views

Adjunctions between Groupoids and Hilbert spaces

I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the ...
7
votes
1answer
141 views

Simplicial nerve functor commutes with opposites

There are two "opposite" functors: $$ op_\Delta\colon sSet\to sSet$$ and $$op_s\colon sCat\to sCat.$$ The first takes a simplicial set to its opposite simplicial set by precomposing with the opposite ...
0
votes
1answer
178 views

To check if a stack is coming from a manifold

Let $\mathcal{D}$ be a stack. An atlas for stack $\mathcal{D}$ is a smooth manifold $X$ and a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$ such that, for any manifold $M$ and a map of ...
3
votes
0answers
199 views

Differential geometry and category theory

Does anyone know of a book/paper/anything, the longer the better introducing differential geometry from a category theoretic point of view? Everywhere it seems categorical language is the elephant in ...
9
votes
2answers
210 views

Balls in Lawvere metric spaces

Let $V$ be the monoidal category $[0,\infty)$ (as a poset) with $+$ and $0$. Lawvere shows that $V$-enriched categories are a more natural generalisation of the notion of a metric space (note no ...
70
votes
5answers
11k views

Has incorrect notation ever led to a mistaken proof?

In mathematics we introduce many different kinds of notation, and sometimes even a single object or construction can be represented by many different notations. To take two very different examples, ...
1
vote
0answers
236 views

Is that correct $\mathbb R^2\cong\mathbb R$ as measurable spaces? [closed]

Is that correct $R^2\cong R$ as measurable spaces? If we consider $R$ and $R^2$ with Borel $\sigma$-algebras, is there measurable map from $R$ to $R^2$ with measurable inverse?
5
votes
0answers
109 views

When is $Ind(C)$ small?

Let $C$ be a small category. Then $Ind(C)$ is the free completion of $C$ under filtered colimits. My sense is that typically, $Ind(C)$ is a large category. But sometimes it is small. For example, if $...
7
votes
1answer
95 views

Bicategory of bimodules over internal monoids

In a monoidal category $(C,\otimes)$ one can consider internal monoids and bimodules over these monoids (provided some additional requirements like existence of coequalizers hold). These monoids ...
8
votes
0answers
92 views

Is every presentable category a limit of locally finitely-presentable categories and finitary left adjoints?

Let $Pr^L$ be the category of presentable categories and left adjoint functors (probably this should be at least a (2,1)-category; anyway ultimately I'm interested in the $(\infty,1)$-setting). For ...
4
votes
1answer
139 views

Exponential law w.r.t. compact-open topology

It is well-known that if a topological space $Y$ is locally compact (not necessarily Hausdorff), then the map $$ \operatorname{Hom}(X \times Y, Z) \to \operatorname{Hom}(X, Z^Y) $$ (here we use the ...
1
vote
0answers
26 views

Domain Monad on Density Operators Using Spectral Order

The spectral order for density operators is given in this paper Coecke Martin 2010. I won't give the full definition here. Essentially, it allows for a partial order of density matrices that forms a ...
2
votes
0answers
71 views

Isomorphism of functors from isomorphism of of their endomorphisms

I am currently stuck at the following problem, and I was hoping that some might know of some literature or known results that might enable me to tackle it. Let $F,G:\mathcal{C}\rightarrow \mathsf{...
11
votes
1answer
276 views

On a surprising property of free theories

Yesterday I observed (and proved) the following odd fact, which I found very surprising. I'm very curious to know if this was known by some people, or if it follows from some other more general fact, ...
2
votes
0answers
114 views

Have you seen this extension property?

I'm looking for a reference for the following extension property: A functor $F: C \to D $ such that for any arrow $f: x \to y$ in $C$ there is an extension $f = \hat{f} \circ i$ with $i:x \...
23
votes
2answers
578 views

Another notion of exactness: how to refine it, and where does it fit?

There are many notions of "exactness" in category theory, algebraic geometry, etc. Here I offer another that generalizes the category of frames, the notion of valuation (from probability theory), and ...
14
votes
1answer
2k views

Why “modding out the homeomorphism” in the category Top makes no rigorous sense?

We can rigorously talk about Top, the category of all topological space, and also FTop, the category of all finite topological space. So I thought, we can define a category FTop', where we “mod out by ...
6
votes
1answer
74 views

Is a weak functor which strictly preserves horizontal composition and which runs between strict bicategories automatically strict?

Let $\mathbf{B}$ and $\mathbf{B'}$ be strict bicategories and $F: \mathbf{B} \to \mathbf{B'}$ a weak functor which preserves horizontal composition strictly (i.e. $Ff * Fg = F(f * g)$ natural in f and ...
10
votes
1answer
424 views

Set-theoretical multiverses and their representation as functors? Why *the* multiverse?

In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see The set-theoretic multiverse) in a category theoretic way, ...
1
vote
0answers
35 views

(Co)Monads with a mixed distributive law on the 2-Category of Groupoids

I am looking for containers on the 2-Category of Groupoids. In particular, though, I would like my container to be both a monad and a comonad with a mixed distributive law. Can someone provide one ...
9
votes
2answers
370 views

What are the reflective subcategories of the category of presentable categories?

I am actually interested in the $\infty$-categorical case, but the same question is meaningful in the $1$-categorical situation as well. A nice property of presentable $\infty$-categories is that if ...
12
votes
2answers
689 views

What is a tensor category?

A monoidal category is a well-defined categorical object abstracting products to the categorical setting. The term tensor category is also used, and seems to mean a monoidal category with more ...
6
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0answers
79 views
+50

Frobenius monads and groupoids

For a while, I was looking for a Frobenius monad on Set. It doesn't exist as pointed out here. I am now looking at the 2-category of groupoids. Does the 2-category of groupoids admit a Frobenius ...
0
votes
0answers
45 views

Comments/references on an obscure category of “rudimentary representations”

Consider a finite toset (cool word) $A=\{a_1<\ldots<a_N\}$. Let an ordered monomial be a finite formal product $b_1\ldots b_M$ with $b_i\in A$ and $b_i\le b_{i+1}$. Consider the following ...
3
votes
1answer
98 views

2-morphisms for Bord(n)

I am currently reading in Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors, and have a (I guess) pretty basic question for my understanding of the (...
4
votes
1answer
73 views

Equivariant non symmetric operads

The definition of a symmetric $G$-Operad is basically a $G$ object in the category of symmetric operads. As far as I understand there is not a good notion of the non symmetric case. I would like to ...
12
votes
0answers
363 views

What's wrong with the obvious argument that the unstable motivic category is an $\infty$-topos?

Fix a base scheme $S$, and let $Sm_S$ be the (ordinary) category of smooth schemes over $S$. Denote by $Psh(Sm_S)$ the $\infty$-category of $\infty$-presheaves on $Sm_S$. Let $Sh_{Nis}(Sm_S)\...
5
votes
0answers
67 views

Products of representables are regular on a regular skeletal Reedy category?

The definition of a (pre)regular skeletal Reedy category in the sense of Cisinski generalizes the intuition behind the Eilenberg-Zilber factorization for simplicial sets, specifically the property ...
6
votes
1answer
199 views

Is there a relationship between norms/transfers in equivariant homotopy theory and norms in the Tate construction / ambidexterity?

In homotopy theory, the word "norm" is commonly used in two different ways (well, surely there are other ways, but these two have a particular familial resemblence). Let $G$ be a finite group. A $G$-...
3
votes
0answers
107 views

Is the simplicial objects functor a comonad?

Let $T$ be the functor of simplical objects $[\Delta^{\mathrm {op}},-]:\mathrm{Cat} \to \mathrm{Cat}$. I am trying to construct counit and comultiplication maps $\eta$ and $\mu$ to make $(T,\eta,\mu)$ ...
4
votes
0answers
71 views

What are morphisms between regularity structures?

In Hairer's notes A Theory of Regularity Structures he defines automorphisms of a regularity structure on page 28. I will recall the definition here: Is there any way of extending this to morphisms ...
-2
votes
1answer
466 views

What is the big picture of algebraic geometry? [closed]

I am trying to understand a big-picture for Algebraic Geometry: Given a category of commutative rings $\mathrm{CRing}$, we can create objects that locally look like these objects called schemes. This ...
4
votes
0answers
74 views

Classification of combinatorial model categories presentable by simplicial presheaves on a Reedy category

Dan Dugger proved that every combinatorial model category can be obtained up to Quillen equivalence as the localization of a model structure on simplicial presheaves on a small category $C$. Is there ...
12
votes
1answer
252 views

What is the coskeleton tower of a quasi-category?

I was giving a talk in a seminar, and I mistakenly said that the coskeleton tower of a quasi-category was its Postnikov tower. Someone corrected me, but a discussion then ensued about what, precisely,...
5
votes
1answer
119 views

Model category of diagrams with the colimit detecting the weak equivalences

Let $I$ be a small category and $\mathcal{K}$ be a combinatorial model category. Is it known a model category structure on the functor category $\mathcal{K}^I$ such that a map of diagrams $D\to ...
6
votes
1answer
196 views

Understanding two proofs in Dwyer and Kan article “Simplicial Localizations”

I can't understand the proofs of propositions 2.6 and 4.2 in https://www3.nd.edu/~wgd/Dvi/SimplicialLocalizations.pdf We have a category $C$ and a family of maps $W$, and we define the ...
12
votes
3answers
559 views

General principles which lead to good questions in many concrete situations [closed]

I believe that in various fields of mathematics there are general principles which might lead to good questions and good results in many concrete situations. I would like to have a list of such ...
15
votes
2answers
291 views

How can I functorially dualise in a symmetric monoidal $(\infty,1)$-category with duals?

If $\mathcal{C}$ is a symmetric monoidal $(\infty,1)$-category with duals, then there should be a functor $$ d: \mathcal{C} \longrightarrow \mathcal{C}^{op} $$ such that $d(x)$ is dual to $x$ for ...
6
votes
3answers
235 views

The skew monoidal structure induced by a functor

$\require{AMScd}$Let $\cal K$ be a 2-category, and $j : A\to B$ one of its 1-cells. Assume that the induced map $$ j^* : {\cal K}(B,B)\to {\cal K}(A,B) $$ precomposing with $j$ has a left adjoint $j_!$...
5
votes
0answers
88 views

References on categorical TVS theory

A survey by Castillo lists results on the category of Banach spaces and on Banach space constructions, such as: Existence of limits in Banach spaces or suitable subcategories Demonstrations of ...
3
votes
1answer
99 views

The binary product of two presentable objects

The binary product of two $\lambda$-presentable objects (in a locally presentable category) is $\mu$-presentable for some regular cardinal $\mu \geq \lambda$ (because all objects are $\mu$-presentable ...