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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6
votes
2answers
198 views

Localizations or quotients of categories?

Motivation: In the classical construction of the derived category of an abelian category, one (roughly) starts with an abelian category $\mathcal{A}$, then considers the quotient category ...
32
votes
2answers
2k views

Category Theoretic Interpretation of Matroids?

Hello everyone! First time poster, long time lurker here. I have a really basic question that has been bugging me for sometime. Specifically, I'm not exactly sure what the 'correct' category ...
4
votes
1answer
67 views

Site dependance of the Cech weak equivalences on simplicial sheaves

Let $\mathcal{T}= sh(C,J)$ a Grothendieck topos of sheaves over a (ordinary) site. One endows the category of simplicial presheaves over $C$ with the "Rezk-Lurie" model structure: that is we start ...
2
votes
0answers
102 views

Functoriality of the adjoint functor construction?

Say we have a category $\mathcal C$, and for every category $\mathcal A$, we have a category $\mathcal D_{\mathcal A}$ and a functor $F_{\mathcal A} : \mathcal D_{\mathcal A} \to \mathcal C$, and, ...
8
votes
1answer
172 views

Non-abelian freeness of SU_2

The distribution of the trace of a random element of $SU_2$ is the Sato-Tate distribution. The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution. ...
6
votes
2answers
133 views

Exactness of an additive left Kan extension

Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair $$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$ where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...
4
votes
1answer
149 views

When does $\mathbf{Top}/X$ embedd fully faithfully into $\mathbf{Top}$?

Under what conditions on the topological space $X$ is the overcategory $\mathbf{Top}/X$ of topological spaces over $X$ equivalent to a full subcategory of $\mathbf{Top}$? Surely if $X$ terminal i.e. a ...
4
votes
0answers
77 views

Limits in Span(Vec)

Let Vec be the category of real vector spaces and linear maps. Let Span(Vec) be the bicategory of correspondences between real vector spaces. I am trying to understand lax limits in Span(Vec). What ...
32
votes
7answers
4k views

What is DAG and what has it to do with the ideas of Voevodsky?

In Toen's and Vezzosi's article From HAG to DAG: derived moduli stacks a kind of definition of DAG is given. I am not an expert and can't see what's the relation between DAG and the motivic cohomology ...
0
votes
0answers
76 views

About stable maps passing through fixed points

In "Notes On Stable Maps and Quantum Cohomology", Fulton and Pandharipande present some results, and their proofs, about the representability of the functor $\mathcal{M}_{g, n}(X, \beta)$, which maps ...
25
votes
6answers
2k views

Synthetic vs. classical differential geometry

To provide context, I'm a differential geometry grad student from a physics background. I know some category theory (at the level of Simmons) and differential and Riemannian geometry (at the level of ...
5
votes
2answers
493 views

Internal equivalence implies weak equivalence for Frechet Lie groupoids?

It is a known theorem that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functors and transformations - ...
21
votes
9answers
7k views

Resources for learning practical category theory

I've been doing functional programming, primarily in OCaml, for a couple years now, and have recently ventured into the land of monads. I'm able to work them now, and understand how to use them, but ...
2
votes
0answers
172 views

What is the right categorical framework for diagonal approximations, cup and cap products and identities between them?

I wanted to know what is the right categorical analog for the diagonal approximation for the singular chain complex $\Delta: C\rightarrow C\otimes C$ as a morphism in the homotopy category of chain ...
5
votes
1answer
260 views

Framed version of braided monoidal category

The operad $\mathcal{D}_2$ of little $2$-disks is an operad whose $n$-th space is a $K(PB_n,1)$ where $PB_n$ denotes the pure braid group on $n$-strands. Algebras over $\mathcal{D}_2$ have a ...
8
votes
1answer
259 views

Infinite dimensional 2-Hilbert spaces

Is there a definition of an infinite dimensional 2-Hilbert space? Finite dimensional 2-Hilbert spaces have been discussed by Baez in http://arxiv.org/abs/q-alg/9609018 In the more recent paper by ...
40
votes
5answers
2k views

A bestiary of topologies on Sch

The category of schemes has a large (and to me, slightly bewildering) number of what seem like different Grothendieck (pre)topologies. Zariski, ok, I get. Etale, that's alright, I think. Nisnevich? ...
2
votes
1answer
194 views

Graphs of tensoring modules

Let $A$ be a ring, and $L,M,N$ an $A^\mathrm{op} \otimes A$-modules. $L \otimes_A M$ is then an $A \otimes A^\mathrm{op}$ module so we can tensor it again with $N$ to get an abelian group $(L ...
-1
votes
0answers
150 views

Kan extensions and special cases

Kan extensions specify the adjoint structures between $\mathbf{Sets^{C^{op}}}$ and $\mathbf{Sets^{D^{op}}}$, where there exists a functor $f:\mathbf{C} \to \mathbf{D}$ and $\mathbf{C}$ and ...
0
votes
0answers
54 views

Complex of “homotopy coherent dg-natural transformations” between dg-functors

Let $\mathcal B$ be a dg-category. I define the dg-category of morphisms $\underline{\mathrm{Mor}}(\mathcal B)$ as follows: objects are triples $(A,B,f)$, where $A,B \in \mathcal B$ and $f \in ...
0
votes
0answers
74 views

graded modules over graded algebra as graded objects

let $\cal{M}$ be the additive symmetric monoidal category of $\mathbb{Z}$-graded $R$-modules (for $R$ commutative ring), and $A$ a commutative algebra in $\cal{M}$. I still think about $Mod(A)$ as ...
12
votes
1answer
273 views

The real numbers object in Sh(Top)

If $X$ is a sober topological space, the real numbers object in the topos $\mathrm{Sh}(X)$ is the sheaf of continuous real-valued functions on $X$. This is proven very explicitly in Theorem VI.8.2 of ...
36
votes
5answers
3k views

Why higher category theory?

This is a soft question. I am an undergrad and is currently seriously considering the field of math I am going into in grad school. (perhaps a little bit late, but it's better late then never.) I ...
2
votes
1answer
136 views

questions of localization of topos

Let $T$ be a topos, and $F \in T$, $T/F$ a localization of $T$. So we have a natural morphism $i: T/F \longrightarrow T$. My questions are: 1.What are the definitions of $i_{\ast}$ and $i^{\ast}$ ...
4
votes
1answer
173 views

Does “simplicial” commute with “Bousfield localization”?

Let $M$ be a model category and $S \subseteq \operatorname{Mor}(M)$ a set of arrows in (the underlying strict category of) $M$. Recall that the left Bousfield localization $L_SM$ of $M$ with respect ...
2
votes
1answer
259 views

Image, kernel, quotient and first isomorphism theorem, in a category of monoid objects

Let $\mathcal{C}$ be a monoidal category and Mon$_{\mathcal{C}}$ the category of monoids (also called algebra objects) on $\mathcal{C}$. Questions: are there definitions of image and kernel for a ...
2
votes
0answers
117 views

Infinite Fubini rule for co/limits

It is a well known fact that given a functor $F\colon I\times J\to C$ then (when everybody exists) $$ \varinjlim_I \varinjlim_J F\cong \varinjlim_J \varinjlim_I F\cong \varinjlim_{I\times J}F $$ Now, ...
7
votes
0answers
97 views

A “small” definition of sub-(∞,1)-topoi

Suppose I have an $(\infty,1)$-topos $\mathcal{X}$ and a (small) set of maps $S$ in $\mathcal{X}$, which therefore generates an accessible localization $S^{-1}\mathcal{X}$. Is there any "small" ...
46
votes
9answers
2k views

What are some examples of interesting uses of the theory of combinatorial species?

This is a question I've asked myself a couple of times before, but its appearance on MO is somewhat motivated by this thread, and sigfpe's comment to Pete Clark's answer. I've often heard it claimed ...
3
votes
2answers
185 views

$ \text{Lan}_KN(-/\mathcal C)\cong N(-/K) $

Let $N(-/\mathcal C)\colon \mathcal C\to \mathbf{sSet}$ be the functor sending $c\in\mathcal C$ to the nerve of the coslice category $c/\mathcal C$. Given a functor $K\colon\mathcal{C}\to ...
32
votes
2answers
2k views

Why do filtered colimits commute with finite limits?

It's not hard to show that this is true in the category Set, and proofs have been written down in many places. But all the ones I know are a bit fiddly. Question 1: is there a soft proof of this ...
12
votes
1answer
739 views

Non-“weakly group theoretical” integral fusion categories?

Is there an integral fusion category of global dimension $210$, such that the simple objects have dimensions $\{1,5,5,5,6,7,7\}$ and the following fusion matrices? $\small{\begin{smallmatrix} 1 & ...
4
votes
2answers
243 views

Topological characterization of injective metric spaces

Let $\ (X\ d)\ \,(Y\ \delta)\ $ be arbitrary metric spaces. A function $\ f:X\rightarrow Y\ $ is called a metric map (with respect to the given metrics $\ d\ \delta$) $\ \Leftarrow:\Rightarrow\ ...
14
votes
3answers
537 views

Characterize the category of rings

(Sub)categories of many well-studied mathematical objects have been characterized purely in terms of their morphisms. Some (famous) examples: Sets and functions, due to Lawvere. Modules over some ...
10
votes
1answer
211 views

A linear category with objects of infinite length but which is otherwise finite?

Fix a ground field $k$. By a linear category I will mean an Abelian category which is compatibly enriched over $k$-vector spaces. A linear category is called finite if it satisfies the following four ...
3
votes
3answers
1k views

Is there a standard notation for binary relations in category theory?

In set theory, I learned that a binary relation is simply a subset of a Cartesian product. Moving on to category theory, it seems that this definition is not enough. Just as a function is no longer ...
1
vote
3answers
379 views

What is the extra property of this sheaf?

I have a particular mathematical structure, and I think it would be enlightening to try to place it in a categorical context. The ...
1
vote
0answers
91 views

Finitely co/continuous monad induced by an operad

It is well known that any operad on a nice monoidal category induces a monad. I was wondering which conditions can be imposed on the operad $T$ (say, in a monoidal closed $\cal V$) so that the ...
22
votes
4answers
2k views

Completion of a category

For a poset $P$ there exists an embedding $y$ into a complete and cocomplet poset $\hat{P}$ of downward closed subsets of $P$. It is easy to verify that the embedding preserves all existing limits and ...
3
votes
1answer
179 views

Do the algebras for a $\infty$-monad form a stable $\infty$-category?

I'm wondering if a monad $T$ on a stable $\infty$-category $\cal C$ has a stable $\infty$-category of algebras, provided $T$ preserves finite limits/colimits. Is this true? Edit: Is something ...
7
votes
1answer
533 views

Higher vector spaces

As far as I know there are different ways to categorify the notion of vector space/module. These appear (for example) when trying to find extended TQFTs. There are at least two ways (presented at ...
2
votes
0answers
589 views

The link between the subfactors and the motives as enriched Galois theories? [closed]

On 29th March 2007, at the "École normale supérieure" of Paris, the mathematician Vaughan Jones, gave a conference (in French) entitled "Les sous-facteurs : une théorie de Galois enrichie" (see here), ...
6
votes
0answers
112 views

Fibrations of the injective model structure on G-simplicial sets

Let $G$ be a discrete group. Consider the category of $G$-simplicial sets endowed with the injective model structure, i.e. cofibrations are the injective maps and weak equivalences are the maps which ...
2
votes
2answers
118 views

$(LLP(Epi), Epi)$ is a WFS on any variety of algebras

This entry in the Joyal catlab claims without proof that in a category $\bf V$ which is a "variety of algebras" the two classes $(LLP(Epi), Epi)$ form a weak factorization system. I interpret this ...
15
votes
1answer
492 views

Categorifications of Zorn's lemma

I'm wondering about categorifications of Zorn's lemma along the following lines. Lemma: if $\mathbf{C}$ is a small category in which every directed diagram of monomorphisms has a cocone of ...
3
votes
1answer
258 views

Comultiplication on Schur functors (& functions). Can it be seen from categorical perspective ?

Consider category of vector spaces. Consider functors from it to itself. They actually form an algebra - since vector spaces can be added and tensor multiplied. Question Is there co-product on this ...
30
votes
4answers
2k views

Cocomplete but not complete abelian category

This is a duplicate of the following question to which I did not receive any answer: http://math.stackexchange.com/questions/238247/complete-but-not-cocomplete-category Let $\mathfrak C$ be an ...
1
vote
1answer
192 views

Relations In Category Theory

Probably a silly question. Suppose that $C$ is a category that does not have finite Cartesian products. So we cannot define a relation on some objects to be a sub object of their Cartesian product (a ...
10
votes
2answers
629 views

If the direct image of f preserves coherent sheaves on noetherian schemes, how to show f is proper?

The other direction is well known. I think it is true and I was told by several other guys doing algebraic geometry that it is indeed true but they did not know how to prove. I am also wondering ...
1
vote
1answer
85 views

Is the extension of a quasi-functor again a quasi-functor?

Let $\mathcal A, \mathcal A', \mathcal B$ be dg-categories over a field $k$ (this assumption allows me not to derive the tensor product, I don't think it is really essential). Let $F : \mathcal A \to ...