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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5
votes
2answers
339 views

A general theory of quasi-functors, generalizing from dg-categories to $\mathcal V$-categories, with $\mathcal V$ monoidal model category

I employ the vast majority of the post to develop the notion of quasi-functor between dg-categories: I think it is important to get the idea. Let $k$ be a field, and let $\mathcal V =\mathbf C(k)$ ...
7
votes
1answer
240 views

The category of categories and adjunctions

What is known about the category that has small categories as objects and adjunctions as morphisms? Obviously, it has neither terminal nor initial objects. But what about other kinds of limits? Are ...
1
vote
1answer
304 views

(Homotopy) limits and colimits in a dg-category

It is known that differential graded categories (or, also, $A_\infty$-categories) are 'incarnations', in some sense, of (stable) $(\infty,1)$-categories. I'm not used to the theory of ...
1
vote
1answer
104 views

Taking zeroth cohomology of a dg-module over a dg-category is “good” on representables, but probably “bad” on cones

This question is possibly related to this other one. Let $\mathcal A$ be a dg-category over a commutative ring $k$. I denote by $\text{dgm-}\mathcal A$ the dg-category of right dg-$\mathcal ...
1
vote
0answers
44 views

Reconstructing an isomorphism of exact triangles in the homotopy cat. of a dg-cat. using a functorially induced isomorphism between cones

This question is strictly related to this other one. Let $\mathcal A$ be a (strongly) pretriangulated dg-category over a field $k$. Let be a diagram in $Z^0(\mathcal A)$, where the rows are ...
3
votes
1answer
106 views

Reconstructing a morphism of exact triangles in the homotopy cat. of a dg-cat. using a “functorial cone map”

I set this problem in the framework of (pretriangulated) dg-categories; everything can probably be translated in the world of stable $(\infty,1)$-categories. Let $\mathcal A$ be a pretriangulated ...
4
votes
2answers
193 views

Serre functor of a subcategory (in particular parabolic category O)

For an additive category $\mathcal C$ there is the notion of a Serre functor on $\mathcal C$, i.e. a an autoequivalence $S$ of $C$ such that there exist isomorphisms $$Hom(A, S(B)) \cong Hom(B, A)^*$$ ...
3
votes
1answer
111 views

weak version of a Baez-Crans 2-vector space?

Baez and Crans defined a 2-vector space to be a category internal to the category of vector spaces (say over the reals). I am interested in categories that are equivalent to Baez-Crans vector spaces ...
2
votes
2answers
202 views

Is antipode unique for bialgebras in arbitrary monoidal categories?

If $B$ is a bialgebra in the category $\tt{Vect}$ of vector spaces (over $\mathbb C$, for example) then $B$ can't have two different antipodes. Is this true for bialgebras in an arbitrary symmetric ...
0
votes
0answers
114 views

Name of Property $t=st \text{ and } s=ts$

What is the name of the property shared by a pair of functions $s,t$ with $$t=st \text{ and } s=ts$$ ( Main example: relation-valued domain and range operations on relations, via ...
3
votes
0answers
88 views

symmetric monoidal dagger endofunctor categories

Take an arbitrary category $C$, and consider $End(C)$, any endofunctor category consisting of some or all endofunctors of $C$. I have several related questions: What restrictions must we impose on ...
1
vote
0answers
106 views

Comonads and the category of Sets

In Vicary's paper, after eq 15, he talks about how the category of internal comonoids $C_\times$ has many properties of the category of sets. We know that a comonad on a category has the same axioms ...
1
vote
0answers
81 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 ...
4
votes
2answers
203 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
1answer
144 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 ...
9
votes
1answer
279 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
2answers
232 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 ...
23
votes
2answers
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 ...
2
votes
1answer
169 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 ...
7
votes
2answers
331 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 ...
1
vote
0answers
77 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
1answer
114 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 ...
5
votes
1answer
136 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 ...
4
votes
1answer
172 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
3answers
596 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 ...
0
votes
2answers
143 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)/
7
votes
2answers
231 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
3answers
614 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 ...
2
votes
1answer
102 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. ...
2
votes
3answers
333 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
1answer
168 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
2answers
133 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 ...
3
votes
1answer
142 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}, ...
1
vote
0answers
76 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
1answer
139 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 ...
0
votes
0answers
157 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
0answers
134 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 ...
2
votes
0answers
208 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 ...
8
votes
0answers
157 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
0answers
142 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
0answers
114 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
0answers
105 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 ...
6
votes
2answers
318 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 ...
7
votes
1answer
175 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 ...
3
votes
0answers
120 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 ...
6
votes
0answers
136 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
2answers
184 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 ...
4
votes
1answer
223 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 ...
2
votes
1answer
186 views

Universal ribbon category of ribbon graphs

I'm skimming through Turaev's "Quantum invariants of knots and 3-manifolds". One of the main results is Theorem 2.5. In my opinion, this Theorem is conceptually half-baked: 1) The ribbon structure on ...
3
votes
1answer
111 views

Strictifying strong monoidal functors

Let $C_1$ and $C_2$ be monoidal categories (not necessarily symmetric or strict) and let $\Psi : C_1 \rightarrow C_2$ be a strong monoidal functor. Is it possibly to construct a strict monoidal ...