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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1answer
80 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
269 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
votes
0answers
193 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
215 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
votes
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
votes
0answers
81 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
177 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 ...
21
votes
1answer
841 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
90 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 that,...
23
votes
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
236 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 t-...
2
votes
1answer
140 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
224 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 A\stackrel{a}{\...
5
votes
0answers
286 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
136 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
166 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 ...
16
votes
1answer
245 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
237 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
108 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
1k 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
142 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 $f^{-1}...
16
votes
2answers
443 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
839 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 ...
3
votes
1answer
169 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
230 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
601 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
206 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
255 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 $\...
7
votes
1answer
380 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
120 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
133 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
192 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
529 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... ...
10
votes
0answers
203 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 $\...
1
vote
0answers
146 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 $\wp(X)\triangleq\...
2
votes
2answers
307 views

In a fibration, where does the generic object live?

In Bart Jacobs. Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics 141. North Holland, 1999. isbn: 9780444508539, the author writes, p. 326: Sometimes, for ...
3
votes
0answers
191 views

Operadic Lift of Lurie's Relative Tensor Product

In Section 4.4 of his book Higher Algebra, Lurie introduces, for a monoid object $A$ of a monoidal quasicategory $C$, and right and left $A$-modules $M,N$, the relative tensor product $M\otimes_AN$. ...
6
votes
0answers
100 views

Finitely presented algebras with isomorphic semilattices of congruences

Let $\mathbb{T}$ be a finitary algebraic theory. For each $\mathbb{T}$-algebra $A$, let $Q (A)$ be the join semilattice of finitely generated congruences on $A$. There is an evident pushforward ...
4
votes
1answer
180 views

Exterior derivative as only (up to multiple) natural operator $\Lambda ^kT^\ast \rightsquigarrow \Lambda ^{k+1}T^\ast$

In Kolar, Michor, & Slovak's book Natural Operations in Differential Geometry, it is proved the exterior derivative is universal in the following sense. Proposition 25.4. For $k>0$ all natural ...
4
votes
2answers
340 views

Natural operators in differential geometry - why are they natural?

I'm reading bits and pieces of Kolar, Michor, & Slovak's Natural Operations in differential Geometry, and I'm having "doubt" about some of the definitions. All I'm trying to do is sheafify some of ...
5
votes
1answer
273 views

Model independent proof of colimit formula for left Kan extensions

I am interested in finding a proof of the colimit formula for left Kan extensions $(\infty,1)$-categories which does not rely on a chosen model. (By a model inpendent proof I mean a proof which uses ...
4
votes
2answers
180 views

Equivalence of natural transformations

Let $\mathcal{C}$ be a small category and $\mathrm{Cat}$ be the 2-category of small categories. Let $F,G : \mathcal{C} \to \mathrm{Cat}$ be two functors and $\theta : F \to G$ be a natural ...
3
votes
1answer
178 views

Definition of Left Operadic Kan Extension for $\infty$-operads

In Lurie's book Higher Algebra, he makes the following definition: Definition 3.1.2.2: Let $M^\otimes\to N(Fin_\ast)\times\Delta^1$ be a correspondence from an $\infty$-operad $A^\otimes$ to another $...
14
votes
1answer
319 views

Does the category of categories-mod-natural-isomorphism have any nonobvious autoequivalences?

The simpler question is to study the 2-groupoid $\mathrm{Aut}(\mathsf{Cat})$ of all autoequivalences of the category of (small) categories (i.e. the groupoid of equivalences of categories $\mathsf{Cat}...
3
votes
0answers
65 views

Bi-adjointness and isomorphisms of (co)limits

Without getting into too many (presumably irrelevant) details, I have a functor $F:A \to B$ between reasonably mellow categories so that for each object $b \in B_0$ the universal morphism $$g_b: \...
4
votes
1answer
204 views

What is the right notion of generalized element of a category?

I've been working out how the internal language of a category C extends to taking the category itself as a type. The most obvious way to interpret $X : \mathbf{C}$ is, of course, that $X$ is an ...
3
votes
0answers
121 views

Topologized category of bounded chain complex

I am reading Segal's paper 'categories and cohomology theories' [1], but there is one claim (in the last example in sec.2) I don't quite understand: Let $\mathcal{C}$ be the category of bounded chain ...
20
votes
1answer
732 views

Lemma 2 from Beilinson's “Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra”, intuition?

This is a followup to here. Consider Lemma 2 from Beilinson's paper "Coherent Sheaves on $\mathbb{P}^n$ and Problems of Linear Algebra", as follows. Lemma 2. For any pair $i$, $j$ such that $0 \...
2
votes
0answers
72 views

How to show these class of morphisms are perfect

In Lurie's book "Higher Topos Theory", a class $\mathsf{W}$ of morphisms in a category $\mathcal{A}$ is called perfect if Every isomorphism belongs to $\mathsf{W}$. it satisfies "2 out of 3 ...