**1**

vote

**0**answers

31 views

### Existence of Colimits in the Definition of Locally Presentable Categories

Basically, my question is simple: why does the definition of a locally presentable category require all colimits exist?
The motivation for this is that I was learning about algebraic posets, and had ...

**3**

votes

**1**answer

62 views

### An orthogonal factorization system on 1-Cob?

Let $1-Cob$ denote the category of oriented 0-manifolds and oriented cobordisms between them. If $W:A\to B$ is a cobordism, i.e. $\partial W\cong A+B$, we write $i^W_{dom}:A\to W$ and $i^W_{cod}:B\to ...

**2**

votes

**1**answer

89 views

### When is an exponential functor a bialgebra?

My question is about the notion of exponential functors as they are frequently defined in the literature on (strict) polynomial functors, e.g., the paper "General Linear and Functor Cohomology" by ...

**0**

votes

**0**answers

81 views

### Tensor product of arbitrary categories [migrated]

I would like to consider a definition of tensor product in the category of (small, finite, whatever is needed) categories, analogous to the tensor product of vector spaces. I will first rewrite ...

**22**

votes

**2**answers

1k views

### Why do Lie algebras pop up, from a categorical point of view?

Groups pop up as automorphism groups in any category.
Rings pop up as endomorphism rings in any additive category.
Is there a similar way to attach a Lie algebra to an object in a category of a ...

**9**

votes

**1**answer

177 views

### Do direct limits (filtered colimits) commute with pullbacks, in C*-algebras?

I asked this question already on math.stackexchange, but maybe it is also useful to ask this here, since it was not answered there.
Suppose we have three directed sequences of $C^*$-algebras, say ...

**6**

votes

**2**answers

195 views

### String diagrams for bimodules over noncommutative algebras?

I'm trying to do some calculations with bimodules (over Azumaya algebras, as it happens), and I need a string diagram notation that mixes the tensor product over the base ring (a symmetric monoidal ...

**3**

votes

**0**answers

103 views

### Categorizing epimorphisms in $\mathscr{L}ex(\mathcal{B}^{Str},\mathsf{Set}_*)$

This is a follow up to this question of mine. The setup: Let $\mathcal{B}$ be an $\mathbb{F}_1$-linear category (Deitmar uses the term Belian); that is, $\mathcal{B}$ is pointed; balanced; contains ...

**9**

votes

**1**answer

393 views

### Is $Lex(\mathcal B,\mathsf{Set}_*)$ an $\mathbb F_1$-linear category?

Following Anton Deitmar, let $\mathcal B$ be an "$\mathbb F_1$-linear category" (Deitmar uses the term "Belian"); i.e., $\mathcal B$ is balanced, pointed, contains finite products, kernels, and ...

**2**

votes

**1**answer

154 views

### Localization of symmetric monoidal category

Let $\mathcal M$ be a symmetric monoidal category, $S\subset \mathcal M$ a collection of objects and morphisms. I would like to construct the localization $\mathcal M \mathop{\longrightarrow}^T ...

**5**

votes

**1**answer

104 views

### Do non-subcanonical Grothendieck topologies always induce a category of fractions?

Suppose that $\mathscr{C}$ is a (possibly higher) category and $J$ is a Grothendieck topology which is not subcanonical. Denote the composite
$$\mathscr{C} \hookrightarrow ...

**19**

votes

**32**answers

976 views

### Formalizations of category theory in proof assistants

What are the existing formalizations of category theory in proof assistants?
I'm primarily interested in public-domain code implementing category theory in a proof assistant (Coq, Agda, ...

**14**

votes

**8**answers

2k views

### References for homotopy colimit

(1) What are some good references for homotopy colimits?
(2) Where can I find a reference for the following concrete construction of a homotopy colimit? Start with a partial ordering, which I will ...

**2**

votes

**0**answers

448 views

### The link between the subfactors and the motives as enriched Galois theories?

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), ...

**4**

votes

**1**answer

178 views

### Constructing a “geometric” model structure on Cat by localizing the “categorical” model structure

Let $\text{Cat}$ be the category of (small) categories and functors. There is a "categorical" (also called "canonical" or "folk") model structure on $\text{Cat}$ in which the weak equivalences are the ...

**3**

votes

**1**answer

139 views

### Profinite completion of a partial order

In Johnstone's Stone Spaces it is proved that the category of profinite partial orders is (equivalent to) the category of ordered Stone spaces (also called Priestley spaces) and that the obvious ...

**5**

votes

**2**answers

441 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 - ...

**1**

vote

**0**answers

69 views

### Characterization of pseudo split epimorphisms in Cat

Is there an easy way to characterize pseudo split epimorphisms in Cat?
Obs: A pseudo split epimorphism in Cat is a functor $F: E\to B $ such that there is a functor $ M: B\to E $ and a natural ...

**4**

votes

**1**answer

86 views

### Good properties of the $H^0$ functor (from quasi-functors to ordinary functors)

Let $\mathcal A, \mathcal B$ be dg-categories over a field $k$. I denote by $\mathcal{RHom}(\mathcal A,\mathcal B)$ the dg-category (defined up to quasi-equivalence) which gives the internal hom in ...

**2**

votes

**0**answers

183 views

### Tensor product of pullbacks of abelian categories

Does the tensor product of abelian categories commute with pullbacks? In more detail:
Let $k$ be a field. We consider $k$-linear small abelian categories ...

**5**

votes

**3**answers

503 views

### opposite category

In the 2-category Cat of small categories, for each category $C$ (an object of Cat) there is also the dual category (I dare not write "dual object") $C^{op}$.
Is ${op}$ the instance in Cat of a more ...

**38**

votes

**10**answers

2k views

### Why is Set, and not Rel, so ubiquitous in mathematics?

The concept of relation in the history of mathematics, either consciously or not, has always been important: think of order relations or equivalence relations.
Why was there the necessity of singling ...

**5**

votes

**2**answers

250 views

### When is/isn't the monoidal unit compact projective?

I am interested in developing intuition for when the monoidal unit in a closed monoidal abelian category is or isn't compact projective. As such, my question is not looking for a yes/no answer, but ...

**0**

votes

**2**answers

125 views

### Smooth Affine algebras are Calabi-Yau

Are all smooth affine algebras over a field Calabi-Yau?
I'm thinking yes since they satisfy Van den Bergh duality with dualizing module themselves (have I made a mistake in this reasoning)/

**6**

votes

**2**answers

201 views

### Pushouts of equivalences of categories

If $f:C\to D$ is an equivalence of categories that is injective on objects, then every pushout of $f$ is also an equivalence. This follows, for instance, because such a functor is an acyclic ...

**10**

votes

**3**answers

531 views

### Are all vector-space valued functors on sets free?

Let $\mathbf{Set}$ be the category of finite sets and functions between them, and let $\mathbf{Vect}$ be the category of finite-dimensional complex vector spaces and linear transformations between ...

**3**

votes

**1**answer

158 views

### internal language for the 2-category of small categories

What is the internal language of the category Cat of small categories?
I found an article by Glynn Winskel and his student Mario Jose Cáccamo about such calculus! However it is limited to a fragment ...

**1**

vote

**2**answers

114 views

### Comonads from monoids

The following construction is probably known. I think it should work in any closed symmetric monoidal category, but I will play it safe and formulate the question in the concrete, cartesian closed ...

**2**

votes

**1**answer

58 views

### Does the category PCM (partial commutative monoids) have a closed symmetric monoidal product?

A partial commutative monoid (PCM) is, roughly speaking, a set with a partially defined binary operation that is as associative as it can be (given that not all products are defined) and commutative. ...

**1**

vote

**3**answers

187 views

### Does the nerve functor (resp. fundamental groupoid functor) preserve homotopy colimits (resp. homotopy limits)?

Let $\pi _1:SS\to Grpd$ denote the fundamental groupoid functor, from simplicial sets to groupoids, and let $N:Grpd\to SS$ denote the nerve functor. Then $\pi _1$ is left adjoint to $N.$
On ...

**3**

votes

**1**answer

117 views

### Exponential objects in a category of abstract automata

I'm working with a more or less standard definition of the category Aut(C) of automata over a category C (where C has finite products) which has tuples $$
A=\langle I_{A},O_{A},S_{A},\sigma_{A}, ...

**17**

votes

**13**answers

5k views

### Programming Languages based on Category Theory

Since some computer scientists use category theory, I was wondering if there are any programming languages that use it extensively.

**2**

votes

**0**answers

107 views

### Localization and Direct limit [migrated]

Let $ A $ be a ring.
Let $ I $ be a preordered set, filtering.
Let $ \Sigma $ a multiplicative subset of $ A $.
Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...

**1**

vote

**0**answers

56 views

### When can I compute the simplicial mapping space from a presheaf to a simplicial presheaf naively?

Suppose that $\mathscr{C}$ is is a small category, $Y$ is a presheaf (of sets) on $\mathscr{C},$ and $X_\bullet$ is a simplicial presheaf. There is a spectrum of simplicial model structures on ...

**3**

votes

**1**answer

119 views

### How to construct a free 2-group on a groupoid?

Let G
be a groupoid. I'm wondering how to construct the free 2-group on G.
By the free 2-group I mean a 2-group $\mathcal{F}\left(G\right)$
equipped with a functor ...

**3**

votes

**0**answers

118 views

### Pursuing an abelian categorical proof of the Zassenhaus Lemma

Fix an abelian category and suppose $M',M,N',N$ are sub-objects of a given object such that $M'\subset M$ and $N'\subset N$. Then there exists a canonical isomorphism
$\frac{M'+(M\bigcap ...

**0**

votes

**0**answers

138 views

### A question about the unbounded derived category of the polynomial ring in infinitely many variables

In this moment I am trying to understand the derived category of the polynomial ring in infinitely many variables over a field $k$, $R=k[x_{1},x_{2},\dots]$ and I am wonder if it is true that ...

**3**

votes

**0**answers

69 views

### Regarding the definition of S-flows over a category (given a monoid S)

(This question was originally directed to Simone Virili, referring to the answer http://mathoverflow.net/a/103840/2926, but could also be addressed to the greater community.)
I was wondering if you ...

**3**

votes

**0**answers

112 views

### When do localizations of presentable (infinity) categories commute?

Suppose that $\mathscr{P}$ is a (locally) presentable $\left(\infty,1\right)$ category (which we can assume WLOG is infinity presheaves on some small $\left(\infty,1\right)$ category) , and $R$ and ...

**5**

votes

**2**answers

1k views

### Generalized Categories for “Higher Homotopy Groupoids”

I was thinking about the definition of higher homotopy groups $\pi_n$ of a topological space in comparison to the common extremely formal fundamental groupoid construction of $\pi_1$. I'd like to be ...

**4**

votes

**0**answers

94 views

### Does simplicial localization with a 3-arrow calculus commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...

**6**

votes

**0**answers

125 views

### Coherent sheaves and Mitchell's embedding theorem

Let $S$ be a scheme. By Freyd-Mitchell's theorem the category of (quasi)coherent sheaves of $\mathcal{O}_S$-modules is equivalent to a full subcategory of the category of left $R$-modules for some ...

**3**

votes

**0**answers

106 views

### When does prolongation preserve sheaves?

Suppose that $(C,J)$ and $(D,K)$ are two Grothendieck (possibly $\infty$-)sites and $f:C \to D$ is a functor such that $$f^*:Psh(D) \to Psh(C)$$ sends sheaves to sheaves. Under what conditions will ...

**1**

vote

**0**answers

95 views

### Retracts of 2-categories

Let $\mathbf{C}$ be a strict $2$-category and let $M = \{(x_\alpha,y_\alpha)\}$ be a collection of object-pairs so that each hom-category $\mathbf{C}(x_\alpha,y_\alpha)$ has an initial element ...

**0**

votes

**3**answers

263 views

### Rank vanishing in tensor categories

Let $\mathcal{C}$ be a cocomplete $\mathbb{Q}$-linear symmetric monoidal category (if convenient, assume that it is locally finitely presentable and/or abelian). Recall that the rank (or dimension) of ...

**6**

votes

**1**answer

154 views

### When does simplicial localization commute with functor categories?

Let $(C,W)$ be a category with a class of weak equivalences, and $D$ a small category. Then I can form the diagram category $(C^D,W^D)$ with objectwise weak equivalences, and its simplicial ...

**5**

votes

**0**answers

103 views

### When does a sheaf of categories represent a homotopy sheaf?

Suppose that $F$ is a sheaf of categories (on a Grothendieck site or even a topological space). By this, I mean a sheaf in the naive 1-categorical sense, so it can equivalently be viewed as a category ...

**-1**

votes

**2**answers

154 views

### Basic category theory: Universality of adjunction unit is justified by Yoneda Proposition in Mac Lane's text [closed]

Background
I am reviewing some category theory, which I did not learn too well the first time around. One text I am using is Mac Lane's. Near the beginning of the chapter on adjunctions (pg 80),
he ...

**3**

votes

**1**answer

200 views

### How do we handle the symmetry condition in nCob and TQFTs?

A $(n+1)$-topological quantum field theory $\mathcal{T}$ is a rigid symmetric monoidal functor from the category $(n+1)$-Cob of $n$-manifolds and $(n+1)$-cobordisms to FdVect.
My question is about the ...

**13**

votes

**6**answers

976 views

### Discrete-compact duality for nonabelian groups

A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for ...