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

6
votes
2answers
238 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 ...
2
votes
2answers
148 views

The source-side-opposite of the arrow category

Given a category $C$, is there a name for the following category: $\mathrm{Obj}(D) = \left\{ (x, y, f) \middle| x, y \in \mathrm{Obj}(C), f \in C(x, y) \right\}$ $D((x, y, f), (x', y', f')) = ...
-1
votes
0answers
47 views

Monoidal product over database table entries

We have seen from Spivak that database schemas are categories, having tables as objects and relationships as morphisms. I am wondering if we can have a monoidal product over objects, that is, over ...
3
votes
1answer
102 views

Seeking more information regarding the “rigoidal category” of $\mathbb{N}$-graded sets

Definitions. By an $\mathbb{N}$-graded set, I mean a set $X$ together with a function $|\Box|_X:\mathbb{N} \leftarrow X,$ called the grading. These will simply be called graded sets hereafter. If ...
2
votes
1answer
122 views

Opposite of an E2-algebra

Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of ...
3
votes
1answer
208 views

About a Double-pseudo-category generalization of the module bicategory construction

To a category with finite limits $\mathscr{C}$, it is associated the bicategory of its spans $Span(\mathscr{C})$. Furthermore the bicategory of (bi)modules (and monoids) on $Span(\mathscr{C})$ is ...
-1
votes
0answers
44 views

Metrics and Measures on a Category of Cats : a cauchy complete category of categories

Is there a suitable way to restrict the functors between objects (and I suppose the objects themselves) of a category of categories such that we have a cauchy complete category. Can we find a complete ...
3
votes
2answers
153 views

Definition of the differential of the Cone of a morphism of complexes [on hold]

Let $(F^\bullet,d_F)$ and $(G^\bullet,d_G)$ be two complexes in an abelian category $\mathbf{A}$. The complex cone $Cone(\varphi)^\bullet$ of a morphism of complexes $\varphi:F^\bullet \to G^\bullet$ ...
5
votes
1answer
243 views

Hahn-Banach theorem for arbitrary locally compact fields?

Does anyone know if the Hahn-Banach theorem is true for every locally compact field? Specifically, let $F$ be a finite algebraic extension of either $Q_p$, the $p$-adic completion of $Q$, or of ...
2
votes
0answers
155 views

About the limitation by size

This could be a big post, so I'll try to summarize my thoughts and divide them into several questions. When working in category theory, I used to choose the following definition. A category $C$ is ...
7
votes
1answer
339 views

What kind of category is a cyclically ordered set?

Background: A preorder is a binary relation $\leq$ which is reflexive and transitive. We can write the transitive property as ${\leq}(a,b)\wedge{\leq}(b,c)\to{\leq}(a,c)$. There are additional axioms ...
-2
votes
0answers
95 views

a naive question: is the category of moniods cartesian closed? Why? [closed]

I read steve awodey's "category theory" and could't solve the exercise in chapter6 above. Here i speak the "category of moniods" the category with objects moniods and arrows homomophisms between ...
6
votes
1answer
81 views

When do powers and ends in functor categories act pointwise?

$\newcommand{\C}{\mathcal C}\newcommand{\I}{\mathcal I}\newcommand{\D}{\mathcal D}\newcommand{\J}{\mathcal J}$Let $(\C, \otimes, I, \multimap)$ be a complete closed monoidal category and $\I$ a small ...
6
votes
1answer
208 views

2-completeness analog of completeness theorem

It's not hard to see that a category is finitely complete if it has finite products and equalizers. In short, this is because one can write all limits as iterations of these two "operations". I ...
3
votes
1answer
258 views

Properties of nerve of strict n-groupoid

The nerve of a groupoid is (by construction) a simplicial set and this works for any category. However there is extra structure in the case of a groupoid. Namely the set of $n$-simplices has an action ...
6
votes
1answer
503 views

permutation of projective limits with inductive limits

Hi everybody, I have a lack of references concerning projective limits and injective limits. Up to my faults in Bourbaki there are only proj and inj limits indexed by a partially ordered set (not a ...
7
votes
2answers
1k views

Is Sheafification Functor Exact?

I know that sheafification functor from the category of abelian presheaves on $C$ to the category of abelian sheaves on $C$. Here, $C$ is a category with Grothendieck pretopology. My question is: ...
5
votes
0answers
399 views

Constructing pointwise Kan extensions as adjoints to some functor

Background I'm working on formalizing some category theory in Coq at https://bitbucket.org/JasonGross/catdb. Currently, I'm in the process of formalizing pointwise Kan extensions. Partly because ...
6
votes
0answers
110 views

Is there a finite test for isomorphisms of rigid monoidal abelian categories, part II

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with an additive tensor functor $Rep_G ...
2
votes
0answers
96 views

DG natural transformation Serre functors

This question might be really easy (or stupid), but I have only vague (heard-about) knowledge of DG categories, so I don't know where to look for an answer. Let $X$ be a smooth projective variety ...
12
votes
1answer
171 views

Is there a finite test for isomorphisms of rigid monoidal abelian categories?

Let $G$ be a semisimple algebraic group. (I'm already interested in the case $G=SL_2$.) Let $\mathcal C$ be a semisimple rigid monoidal abelian category endowed with pair of exact tensor functors ...
27
votes
35answers
2k 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, ...
0
votes
1answer
193 views

Order-Perserving Bijection $f:A\to A^*$?

Let $A$ be a well-quasi-ordered infnite set. Does there exist an order-preserving bijection $f:A\to A^*$, where $A^*$ is the free monoid over $A$ under the subword ordering? Would this subword ...
3
votes
2answers
178 views

How to simplify the proof of right-properness?

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback ...
4
votes
1answer
92 views

Is this additive equivalence a triangulated equivalence?

Let $\mathcal{C}$ and $\mathcal{D}$ be triangulated categories, and suppose that there exists an additive equivalence $F: \mathcal{C} \to \mathcal{D}$. Suppose further that $\mathcal{C} = ...
0
votes
1answer
54 views

Mapping an arrow from the direct limit of a diagram to the family of arrows from the diagram

Consider the direct limit of an indexed family $\{a_n\}_{n\in \omega}$: $\require{AMScd}$ \begin{CD} a_0 @>>> \ldots @>>> a_n @>>> a_{n+1} @>>> ...
4
votes
0answers
88 views

Ref request: modelling regular theories as an injectivity condition

The background to the following observation is standard material in categorical logic, and I thought this was was too — I don’t remember learning it, but I don’t think it is original — but I can’t now ...
27
votes
2answers
4k views

Is there a category structure one can place on measure spaces so that category-theoretic products exist?

The usual category of measure spaces consists of objects $(X, \mathcal{B}_X, \mu_X)$, where $X$ is a space, $\mathcal{B}_X$ is a $\sigma$-algebra on $X$, and $\mu_X$ is a measure on $X$, and measure ...
2
votes
0answers
42 views

Holonomy 2-functor transformation by transition functions

The holonomy 2-functor on a $\mathcal{G}$-principal 2-bundle associates a bigon: $$\mathsf{hol}_i(\Sigma):\mathsf{hol}_i(\gamma)\Rightarrow \mathsf{hol}_i(\gamma')$$ in $\mathcal{G}$ to each bigon: ...
2
votes
0answers
71 views

Factorization system in a derivator

Has anybody attempted to define the notion of a factorization system "in" a derivator? Something on the lines of this: let $\mathbb{D}$ be a (strong) derivator. We define the orthogonality relation ...
6
votes
1answer
173 views

What is the right adjoint of the tensor product in a closed monoidal functor category?

The nLab says the following about closed monoidal functor categories: Let $C$ be a complete closed monoidal category and $I$ any small category. Then the functor category $[I, C]$ is closed ...
0
votes
0answers
60 views

Is there a general way to define invariants in a category, using generalized elements?

In Awodey's category theory, page 37, he uses a specific generalized elements $2 \to X$ and $2 \to A$ for posets $A=\{a \leq b \leq c, \}$ and $X=\{x \leq y, x \leq z, \}$to make an invariant in the ...
8
votes
1answer
240 views

Correspondences as generalized morphism between $C^*$-algebras

While reading about Morita equivalence in the category of $C^*$-algebras I met also the following notion: a correspondence between two $C^*$-algebras $A,B$ is a pair $(X,\varphi)$ where $X$ is a ...
5
votes
1answer
200 views

Does this notion related to species/operads/FI-modules have a name?

Let $B$ be the symmetric monoidal category of finite sets and bijections with disjoint union. Let $C$ be a symmetric monoidal category. Is there a standard name for a lax monoidal functor $F:B \to C$? ...
21
votes
9answers
2k views

What is… A Grossone?

Y. Sergeyev developed a positional system for representing infinite numbers using a basic unit called a "grossone", as well as what he calls an "infinity computer". The mathematical value of this ...
7
votes
3answers
1k views

A categorical method to, say, determine the cardinality of a group

I am trying to figure out how much one can figure out about an object using category theory. Ideally, any property that is well defined up to isomorphism should be computable using only category ...
1
vote
1answer
115 views

How do you rigidify a Bousfield localization?

I'm learning about Bousfield localizations. For a triangulated category satisfying some axioms, a Bousfield localizations can be described as an idempotent functor $L:D \to D$. I thought there is a ...
8
votes
1answer
268 views

Non-Cartesian Monoidal Model Structure on a Slice Category

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...
1
vote
0answers
157 views

Reference request: Models of isomorphic languages result into isomorphic categories

This is basically a reference request by someone who has not been educated as a logician and would like to be rigorous about certain preliminary aspects of model theory. Fix an uncountable universe ...
5
votes
2answers
140 views

How to make a premodular category a modular tensor category?

A premodular category (also called ribbon fusion category) is roughly speaking a tensor category where fusion and braiding of the objects are defined. With an extra nondegeneracy condition for the ...
5
votes
2answers
518 views

Isomorphism in category of finite automata

What does meanthat two finite automata is equivalent? I think that we must define category of finite automata, i.e. we must define $\mathrm{Hom}(A,B)$, where $A,B$ be an arbitrary finite automata. ...
5
votes
1answer
119 views

Do $RHom(C,D)$ and $DG(C,D)$ have equivalent homotopy categories?

Toen in The homotopy theory of dg-categories and derived Morita theory Section 6 introduced the internal Hom's between dg-categories. Actually for two dg-categories $C$ and $D$, Toen defined $$ ...
3
votes
1answer
146 views

Action of a strict 2-group on a category gives autoequivalences?

A strict action of a strict 2-group (seen as 2-category with only one object $\star$) $\mathcal{G}$ on a 2-space (principally a category) $\mathcal{M}$ is a strict 2-functor $\Phi:\mathcal{G}\to ...
29
votes
2answers
852 views

Cantor's theorem for presheaves?

Some years back (before MathOverflow was born), Tom Leinster asked an interesting question at the $n$-Category Café which I don't recall ever seeing an answer for: Does there exist a ...
5
votes
1answer
496 views

Projectives and Injectives in Functor Categories

Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories? Here is a precise question. Let $C$ be a small category, whose ...
2
votes
0answers
88 views

Factorization of a map through a square

Assume to have an abelian category $\mathcal{A}$, and consider its derived category $\mathcal{D^b(A)}$). Let $F:\mathcal{D^b(A)}\rightarrow\mathcal{D^b(A)} $ be a functor between triangulated ...
0
votes
1answer
128 views

colimits in Cat via coproducts and coequalizers

I am attempting to do a calculation of a colimit in $Cat$, the category of small categories. To this end, people have suggested that I do this by calculating coproducts and using coequalizers. I ...
6
votes
3answers
1k views

Crossed module structure on homotopy groups.

A crossed module is a pair of groups $C$ and $G$, an action of $G$ on $C$, and a homomorphism $\partial: C \to G$ that satisfy $\partial(g\cdot c)=g(\partial c)g^{-1}$, and $cc'c=(\partial c)\cdot ...
2
votes
1answer
253 views

Groupoid isomorphism vs. group isomorphism

Assume that $\Gamma$ is a group with neutral element $e$. We associate to $\Gamma$ the following groupoid $G$: $G=\Gamma \times \Gamma,\;\;\;G^{(0)}=\Gamma \times \{0\},\;\;s(a,b)=(a,e),\;\;\; ...
5
votes
3answers
215 views

strict 2-groups VS crossed modules

nLab defines a strict 2-groups in many different but equivalent ways, among them: an internal group object in Cat, an internal group object in Grpd Also, it is known that strict 2-groups may be ...