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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2
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91 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
156 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 ...
2
votes
1answer
274 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),\;\;\; ...
1
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0answers
61 views

Non-trivial but simple concrete examples for some categories related to Tensor/Fusion categories

I'm writing a note on Tensor and Fusion categories, the readers of which are physicists rather than mathematicians. So instead of giving abstract definitions I have to give examples to inspire each ...
4
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0answers
109 views

What structure of a monoidal simplicial model category is preserved by taking the opposite category?

Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, ...
5
votes
2answers
172 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
3answers
233 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 ...
0
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1answer
136 views

Mac Lane strictness theorem and categorifiability of fusion rings

The Mac Lane strictness theorem states that any monoidal category is monoidally equivalent to a strict monoidal category (see here section 2.8). Q1: Is it true that any fusion category is monoidally ...
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33 views

going from basic category theory to 2-category theory [duplicate]

It seems to me that 2-category is the natural frame to express many of the interesting concepts in computer science : monoidal categories and the various monoids (inc monades etc..) end and coend as ...
2
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0answers
85 views

Are there any detailed references for the enriched yoneda lemma?

I am just starting out learning enriched category theory, and I am looking for a reference proof of the Yoneda lemma for categories enriched in a monoidal category. Thank you for your help.
7
votes
2answers
223 views

What is the “quaternionic” super Brauer group?

In addition to the two reasonably well-known categories $\mathrm{SuperVect}_{\mathbb R}$ and $\mathrm{SuperVect}_{\mathbb C}$ of real and complex super vector spaces, each of which is monoidally ...
3
votes
2answers
186 views

Does every bicategory have a “delaxing object”?

If I'm not mistaken, there is a bicategory $\mathsf{Monad}$ given as follows: Start with the associative operad. Deloop it to obtain a multicategory. Adjoin objects and morphisms as necessary to ...
5
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1answer
158 views

Directed homotopy in the Cayley graph of a monoid

There is a the notion of the Cayley graph $C(G)$ of a group $G$ (which depends on a given presentation $G \cong \mathcal F(S) / \sigma$ where $\mathcal F$ is the free group functor and $\sigma$ some ...
5
votes
0answers
177 views

reference request: category of crystals on a scheme is locally noetherian

I'm looking for a reference to the fact that the category of crystals on any scheme is locally noetherian. This is stated in Gaitsgory's paper "Crystals and D-modules" but he doesn't provide a ...
2
votes
1answer
78 views

What do you get when you apply a universal cocone to a colimit functor

Any colimit can be represented as a functor $F$ left adjoint to a particular diagonal functor $\Delta: C \rightarrow C^J$. The unit of this adjunction is the natural transformation $\eta_K: K ...
6
votes
1answer
257 views

Jets in synthetic differential geometry

As I understand it from Kostecki's notes, the $k$-jet $j^k f$ of a function $f: R^n \to R^m$ should be the map $$f^{D_k(n)} : {(R^n)}^{D_k(n)} \to (R^m)^{D_k(n)},$$ where $$D_k(n) = \{(x_1, \ldots, ...
7
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0answers
183 views

A definition of the homotopy colimit of a coherent diagram

Suppose I am given a homotopy coherent diagram of spaces of shape $I$ (This is a simplicial functor $F:\mathfrak{C}[I] \to Top$, where $\mathfrak{C}$ is the standard cofibrant replacement functor in ...
2
votes
2answers
198 views

Question about Enriched Categories and Functors

How would one describe the process of enriching a category C over some monoidal category D? Is there some functor between them that adds structure to the hom-sets?
2
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1answer
94 views

A question on 2-bundles

In this paper, the authors +John Baez and +Urs Schreiber defined (page 15) "transition functions" for a special kind of 2-bundles (those whose the base space is a ordinary smooth space augmented to a ...
0
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0answers
80 views

Does a left Kan lift of a homset functor Hom(*, - ): C -> {Set} through the forgetful functor {M-Set} -> {Set} exist?

Let $M$ be a monoid and let $\mathsf{U} : M\mathsf{Set} \to \mathsf{Set}$ denote the functor which forgets the action on an $M$-set. Given a category $C$ with a distinguished object $p$ as baspoint, ...
2
votes
0answers
174 views

Algebraic K theory, Karoubi completion and splitting

Suppose $\mathcal{C}\subset \mathcal{C}'$ is a pair of pre-triangulated smooth DG categories over a characteristic-zero basefield (say, $\mathbb{C}$), such that $\mathcal{C}$ is faithfully embedded in ...
19
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1answer
629 views

Variant of Conceptual Completeness

Let $\mathcal{C}$ and $\mathcal{D}$ be pretopoi, and let $f: \mathcal{C} \rightarrow \mathcal{D}$ be a pretopos functor (that is, a functor which preserves finite coproducts, finite limits, and ...
3
votes
0answers
79 views

What are universal abstract $\sigma$-algebras on $\sigma$-frames?

Originally asked on MSE. In this paper, the authors make the following definitions: An (abstract) $\sigma$-algebra is a boolean algebra with countable joins. A $\sigma$-frame is a bounded lattice ...
1
vote
0answers
59 views

Determining finite abelian groups among algebraic theories by counting

Let $T$ be a Lawvere theory (algebraic theory) which contains the theory $T_{\mathrm{grp}}$ of groups as a subtheory (so that $T$ has, in general, more equations that $T_\mathrm{grp}$). Suppose ...
22
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9answers
3k 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 ...
3
votes
1answer
232 views

Reference for t-structures on stable model categories

What kind of definitions of t-structures on stable model categories have been investigated in the literature? Of course, one can always define a t-structure on a stable model category as a ...
2
votes
1answer
139 views

Additive functors to abelian groups: “additional structure” and functors induced by “additive pseudo-functors”; references?

Let $\mathcal{A}$ be a small additive category. Consider the category $PreSh(\mathcal{A})$ of all additive functors from $\mathcal{A}^{op}$ into abelian groups; note that this category is abelian and ...
5
votes
0answers
218 views

Do differential objects form triangulated categories?

Let $\mathcal{A}$ be a (fixed) additive category. To a differential object $(A,a)$ for $\mathcal{A}$ (so, $a:A\to A$ and $a^2=0$) one may associate an $\mathcal{A}$-complex $\dots \to ...
5
votes
0answers
275 views

What is an Elementary “Homotopy, Model” Topos?

Context: Def (Rezk): A (Grothendieck) homotopy topos is a homotopy left exact Bousfield localization of the model category of simplicial presheaves sPsh(C) on a small simplicial category C. Thm ...
3
votes
0answers
127 views

Test categories applied to Dold-Kan correspondence?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to ...
8
votes
1answer
159 views

Is the modularisation of a unitary fusion category always unitary?

Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières ...
15
votes
1answer
231 views

Reference request: Morita bicategory

I have two closely related questions: Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners? I've heard this bicategory called the ...
4
votes
1answer
233 views

What kinds of limits does localization of commutative rings reflect?

Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of ...
3
votes
1answer
105 views

Category enriched over a monoidal 2-category

Consider a monoidal 2-category (or bicategory) B. For example, B could by the 2-category (finite sets, finite correspondences, isomorphisms of correspondences) with monoidal structure given by ...
22
votes
2answers
999 views

“Why” are monadicity and descent related?

This question is probably too vague for experts, but I really don't know how to avoid it. I've read in several places that under mild conditions, a morphism is an effective descent morphism iff the ...
8
votes
0answers
139 views

Which nice subcategories of $\mathsf{Top}$ are locally cartesian closed?

For a class $\mathcal{J}$ of topological spaces, let $\mathsf{Top}_\mathcal{J}$ denote the category of $\mathcal{J}$-generated spaces, i.e. those spaces $X$ such that $U\subseteq X$ is open iff ...
16
votes
2answers
432 views

What categorical property of monoidal categories picks out the ones with duals?

Recall that a monoidal category $\mathcal C$ is rigid if every object $X\in \mathcal C$ has both left and right duals, i.e. objects $X^l$ and $X^r$ with maps $X^l \otimes X \to \mathbf 1 \to X \otimes ...
21
votes
1answer
821 views

Topology of categories, very basic facts surrounding Quillen's Higher Algebraic K-Theory I

In his paper Higher Algebraic K-Theory I (see [here][1]), Quillen introduces a topological space $BC$, called the classifying space of $C$, and tries to relate its topology to the categorical ...
2
votes
1answer
161 views

Faithful exact functors to tensor categories

Let $P$ be a "nice" $k$-linear abelian tensor category (e.g. A tannakian category or a fusion category over a field $k$) and $F: M\to P $ an additive $k$-linear exact and faithful functor. I want to ...
6
votes
0answers
221 views

Many-sorted nominal sets as sheaves

The category of nominal sets can also be presented as the Schanuel topos, the category of sheaves on $\mathbb{I}^\mathsf{op}$ under the atomic topology, where $\mathbb{I}$ is the category of finite ...
15
votes
1answer
588 views

Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
3
votes
0answers
181 views

Defining Inertia Stack

Let $X$ be a topos and $F: \zeta \rightarrow X$ a stack on $X$. Now in the paper http://arxiv.org/pdf/math/0411337v2.pdf ( Definitions 2.1.1.5 and 2.1.1.1) of Lieblich, he describes the inertia stack ...
6
votes
1answer
250 views

What is known about the large cardinal strength of Shelah's categoricity conjecture?

Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of ...
6
votes
1answer
360 views

P.G.Goerss, J.F.Jardine, “Simplicial Homotopy Theory” prerequisites

I know that such questions may be better suited for math.stackexchange, but I believe that that the topic of simplicial homotopy theory is advanced enough for mathoverflow. Besides, I know that there ...
4
votes
0answers
113 views

Topos Theory, internal Heyting Algebra

Given a topos $\mathcal{E}$ with subobject classifier $\Omega$. If we denote by $N\Omega$ the former of all local operators on $\Omega$, that is, Lawvere–Tierney topologies of $\mathcal{E}$, it is ...
3
votes
1answer
121 views

What is an example of a colimit-dense generator which is not dense?

An object $G$ of a category $\mathcal{C}$ is a dense generator if every object $X$ is the colimit of the canonical diagram of copies of $G$ mapping to $X$. (This canonical diagram is indexed by the ...
3
votes
1answer
156 views

Left adjoint of pullback

In Awodey's Category Theory, p233 of 2nd ed. (or p205 of 1st ed.), he states: Indeed, the UMP of pullbacks essentially states that composition along any function α is left adjoint to pullback ...
10
votes
1answer
509 views

Categorical or simplicial introduction to modern homotopy theory

I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering... ...
9
votes
0answers
198 views

$\mathcal{M}(\mathcal{D}_X)$ and $\mathcal{M}^r(\mathcal{D}_X)$ have natural tensor category structures?

Write $\mathcal{M}^\ell(\mathcal{D}_{X/S}) = \mathcal{M}(\mathcal{D}_{X/S})$ for the category of left $\mathcal{D}$-modules over $X$ and $\mathcal{M}^r(\mathcal{D}_{X/S})$ for the category of right ...
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139 views

Coinduction in a presheaf topos?

Fix a presheaf topos $\mathcal{E}\triangleq \mathbf{Set}^{\mathcal{C}^\mathsf{op}}$. Then, for any object/presheaf $X:\mathcal{E}$, I can define the presheaf of relations on $X$ as ...