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

learn more… | top users | synonyms

2
votes
0answers
25 views

6j symbols with Majorana indices

The Levin-Wen model is a Hamiltonian formulation of Turaev-Viro (2 + 1)d TQFTs. It can be constructed from a unitary fusion category $\mathcal{C}$, which can be equivalently defined using $6j$ ...
-3
votes
0answers
108 views

A question about Category Theory [on hold]

The Review of Symbolic Logic for June 2015 contains an article by Michael Ernst, in which it is proved that Unlimited Category Theory (as defined by S. Feferman) is inconsistent. This seems to me to ...
2
votes
0answers
23 views

Does the $D$-property have universal objects?

A space $(X,\tau)$ is called a $D$-space if whenever one is given a neighborhood $N(x)$ of $x$ for each $x\in X$, then there is a closed discrete subset $D\subseteq X$ such that $\{N(x): x\in D\}$ ...
2
votes
0answers
92 views

Can such categorical notion of action be formalized?

I'm wondering if it's possible to find an universal construction for a general concept of action for (single-sorted?) finite product sketches, such that one of those is "acting" on the second in the ...
6
votes
3answers
524 views

Category theory for Algebraic Geometry

How much of category theory should I know to view schemes, sheaves and cohomology concepts as concrete cases of abstract categorical concepts? Is there a textbook of category theory for AG people?
1
vote
0answers
79 views

Category-theoretic characterization of zero-dimensional spaces

Some background: a zero-dimensional space is one admitting a basis of clopen sets, whereas an extremely disconnected space is one where the closures of open sets are open. In the category CHauss of ...
4
votes
1answer
105 views

Kan extensions of pseudofunctors

Can anyone suggest a reference for (left) Kan extensions of pseudofunctors? In particular, say we are given bicategories $\mathscr{A,B,C}$ and pseudo functors $\mathscr A \xrightarrow{G} \mathscr ...
14
votes
3answers
353 views

Brouwer's theorem for the Cauchy reals

Brouwer famously proved, using principles motivated by intuitionistic choice sequences, that every function $\mathbb{R}\to \mathbb{R}$ is continuous. In Sheaves in geometry and logic (section VI.9), ...
16
votes
0answers
511 views

A Linear Order from AP Calculus

In teaching my calculus students about limits and function domination, we ran into the class of functions $$\Theta=\{x^\alpha (\ln{x})^\beta\}_{(\alpha,\beta)\in\mathbb{R}^2}$$ Suppose we say that ...
4
votes
1answer
196 views

The “$\infty$”-column in the periodic table of n-categories

A monoid is the same as a category with a single object. A monoidal category is the same as a bi-category with a single object. A commutative monoid is the same as a bi-category with a single object ...
12
votes
1answer
249 views
+50

Which sequential colimits commute with pullbacks in the category of topological spaces?

This question was asked on math.stackexchange.com without a reaction. Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ ...
3
votes
0answers
56 views

Natural transformations of $A_\infty$-functors (between dg-categories) are “directed homotopies” (reference?)

Let $\mathbf A$ and $\mathbf B$ be dg-categories over a field, viewed as $A_\infty$-categories. The $A_\infty$-category (actually, dg-category) of strictly unital $A_\infty$-functors $\mathbf A \to ...
0
votes
0answers
49 views

Generating a series representation for the inverse of the operator $f(f)$

I was considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...
3
votes
0answers
88 views

Does this notion of “$\mathcal{F}$-digraph” appear in the literature?

By a digraph, I mean a quiver with no multiple edges. So in particular: Loops are okay. An infinite set of vertexes is okay. Furthermore, I will tend to identify each digraph with its underlying ...
4
votes
2answers
255 views

Algebras for probability monad

What is the Eilenberg-Moore category for the non-finitary probability distribution monad is, that is, the monad $D \colon \mathbf{Set} \to \mathbf{Set}$ defined by $$ DX = \left\{ p \in [0,1]^X \ ...
-1
votes
1answer
70 views

Categorical product of graphs and chromatic number

Let $(G_i)_{i\in I}$ denote a family of simple, undirected graphs (finite or infinite). Let $\prod_{i\in I}G_i$ denote their categorical product. Why do we have the inequality $$\chi(\prod_{i\in ...
2
votes
0answers
95 views

About a (new?) definition of transformation (anti.transformation) as a link between natural and dinatural transformations

This is not a hard topic, but I post here as "reference request" or because elementary aspects (but not previously vocalized) can be interesting too for researchers. Given $F: \mathscr{A}\to ...
2
votes
1answer
78 views

Model structures on diagrams indexed by a Reedy category

I'm interested in the way to put a model structure on the category of functors $F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the ...
10
votes
2answers
528 views

What's a good introduction to category theory for someone doing analysis?

I do functional analysis, and diagrams are popping all over the place. It is about time I learned me some category theory. Any recommendations?
2
votes
0answers
94 views

Generalizing disjointness

The following definition generalizes set-theoretic disjointess: Definition 0. (Autonomy). Given a Lawvere theory $\mathsf{T}$, a $\mathsf{T}$-algebra $X$, and an indexed family $S$ of subalgebras ...
1
vote
1answer
84 views

How to define the internal hom between presheaves valued in cotensored categories?

First let $\mathcal{V}$ be a closed symmetric monoidal category and $\mathcal{M}$ be a category enriched over $\mathcal{V}$. Moreover we assume $\mathcal{M}$ is cotensored, or powered over ...
4
votes
0answers
142 views

Schwede-Shipley theorem for monoidal categories?

The Schwede-Shipley theorem gives a criterion for a presentable stable $\infty$-category to be the category of modules over an $\mathcal{E}_1$-algebra. Is there any similar criterion for a monoidal ...
0
votes
0answers
56 views

A construction on lax.functor

Consider for simplicity only locally small 2-categories. Given a 2-category $\mathscr{A}$ let $|\mathscr{A}|$ its 2-graph (forget the horizontal composition). Given a 2-graph $\mathcal{G}$ let ...
9
votes
1answer
256 views

Adding inverses to a symmetric monoidal category (Reference?)

As we all know, the forgetful functor $\mathsf{Ab} \to \mathsf{CMon}$ from abelian groups to commutative monoids has a left adjoint, the Grothendieck group. I would like to categorify this ...
33
votes
2answers
789 views

The formal p-adic numbers

The real numbers can be defined in two ways (well, more than two, but let's stick to these for now): as the Cauchy completion of the metric space $\mathbb{Q}$ with its usual absolute value, or as the ...
15
votes
1answer
317 views

The Gelfand duality for pro-$C^*$-algebras

The Gelfand duality says that $$X\to C(X)$$ is a contravariant equivalence between the category of compact Hausdorff spaces and continuous maps and the category of commutative unital $C^*$-algebras ...
3
votes
0answers
86 views

If $C$ has all geometric realizations of simplicial objects, what other colimits does it have?

Let $C$ be an $\infty$-category. Suppose that every diagram $\Delta^{\mathit{op}} \to C$ has a colimit. Is there any characterization of small categories $I$ such that every diagram $I \to C$ has a ...
0
votes
0answers
45 views

About the functors composition completeness

If $F: \mathscr{A}\to \mathscr{B}$ is a functor and $[\mathscr{C}, \mathscr{D}]$ is the category of functors and natural transformations between two given categories $\mathscr{C}$ and $\mathscr{D}$ ...
1
vote
0answers
115 views

History of categorical localization sans calculi of fractions

This question arises from a paper which I've just found and skimmed: FW Bauer, J Dugundji. Categorical homotopy and fibrations. Transactions of the American Mathematical Society, 1969 With 28 ...
4
votes
2answers
272 views

Categories of finite objects

In my experience, category theory is very successful at providing powerful machinery to reason about large objects or objects unrestricted in size, for example (logical) models (via accessible ...
2
votes
2answers
338 views

Reference for higher categorical analogue of algebraic cycle? [closed]

Are there higher categorical analogues of algebraic cycles? What are some references? This question arise in an attempt to generalize algebraic cycles towards higher dimensional algebra. Has there ...
1
vote
1answer
90 views

Can hypercompletion be an essential localization?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory, i.e. its inclusion functor has a left-exact left ...
4
votes
1answer
139 views

locally noetherian categories and the category of quasi-coherent sheaves over a noetherian scheme

It is known that a ring $R$ is noetherian if and only if direct sums of injective $R$-modules are injective if and only if every injective $R$-module is a direct sum of indecomposable injective ...
6
votes
1answer
245 views

Can hypercomplete objects be coreflective?

The subcategory of hypercomplete objects in an ∞-topos is a left-exact-reflective subcategory by the remarks after 6.5.2.8 of Higher topos theory. Can it ever happen that this subcategory is ...
3
votes
1answer
57 views

Semisimple monoidal category with duals

We say that an abelian category is semisimple if every object is a semisimple object, which is to say, a direct sum of finitely many simple objects. Let $({\cal C},\otimes,*)$ be a semisimple ...
3
votes
1answer
137 views

Is a pullback along a Dold fibration a homotopy pullback?

Let $$ \begin{array}{ccc} A & \to & B \cr\downarrow&&\downarrow \cr A'& \to &B' \end{array} $$ be a pullback square in the category of all topological spaces (not just in a ...
3
votes
1answer
111 views

About a closed strucure on profunctors

Let $Prof$ the bicategory with profunctors (on small categories), arrows are like $D: \mathscr{A} \dashrightarrow \mathscr{B}$ and this means that $D: \mathscr{A}^{op} \times \mathscr{B}\to Set$. ...
8
votes
1answer
208 views

Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...
2
votes
1answer
96 views

Do the full subcategories have a simple structure in higher category theory?

Let $C \in Cat$ be an $(\infty,1)$-category. Let $P$ be the partially ordered subset of full subcategories of $C$. Is there a (canonical?) functor from the nerve of $P$ to $Cat$? I think the answer ...
8
votes
1answer
114 views

Detecting positive endomaps of the formal reals

A locale is a sort of "formal topological space", which "may not have enough points to separate its open sets". For instance, there is a "locale of all real numbers that are both rational and ...
0
votes
1answer
98 views

Adjointable Abelian Monoidal Functor

Given two abelian monoidal categories ${\cal C,D}$ (where the monoidal operation is bilinear) and an additive monoidal functor $F:{\cal C} \to {\cal D}$. Will $F$ always admit an adjoint?
2
votes
1answer
102 views

Does the following characterize local presentability?

Let $\mathcal C$ be a cocomplete category. Consider the following two conditions: $\mathcal C$ is locally presentable. The Yoneda embedding $$\mathcal C \hookrightarrow \{\text{continuous functors ...
9
votes
1answer
179 views

Does every Lawvere theory arise in this way?

By a Lawvere theory, I mean a finite-product category $\mathsf{T}$ equipped with a distinguished object, such that every object of $\mathsf{T}$ can be expressed as a finite product of the ...
4
votes
1answer
174 views

An example for a construction on monads/operads?

Suppose that $C$ is a bicategory. (I only need a monoidal category, i.e. one object bicategory, but I will stick with bicategories, since theory of monads is more commonly stated in that setting). A ...
5
votes
0answers
138 views

“Generalized theory of polynomials” for a given commutative Lawvere Theory

I am trying to understand Nikolai Durov's "New Approach to Arakelov Geometry" right now and it got me thinking about a particular thing. Let $R$ be a commutative, associative ring with unit. We can ...
2
votes
2answers
168 views

When is the adjoint to a monoidal functor monoidal?

Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F : \mathcal C \to \mathcal D$ is lax monoidal if it is equipped with maps $1_{\mathcal D} \to F(1_{\mathcal C})$ and $F(X) ...
3
votes
0answers
99 views

What about “bilax” functors?

in [G] p.29, J.W Gray define the 2-comma category $[F, G]$ of two 2-functors $F: \mathcal{A}\to \mathcal{D},\ G: \mathcal{B}\to \mathcal{D}$. This definition work well also if we suppose $F$ a ...
3
votes
0answers
226 views

Quasicategorical Construction of a Cosimplicial Map of Rognes

In John Rognes' Galois theory monograph he constructs something called the Hopf-cobar complex for a coalgebra object $H$ (in spectra) and a comodule algebra $X$. It is, intuitively, the object whose ...
2
votes
1answer
97 views

Exponential locales and a pointless version of the compact-open topology?

TL;DR: compact-open topology for Homs of locales? Let $\mathcal{L}$ be a full subcategory of the category $\mathcal{Loc}$ of locales. For two locales, $A$ and $B$, is there a nice way to make an ...
0
votes
1answer
87 views

Yetter-Drinfeld modules as rigid category

I'm reading a proof of the following theorem If $H$ is a Hopf algebra with invertible antipode then Yetter-Drinfeld modules of finite dimension form a rigid category. In this proof we define ...