User buschi sergio - MathOverflow

About a General Definition of Profunctor

2012-01-03T15:40:04Z

Given categories $\mathcal{A},\ \mathcal{B}$ let $\mathcal{A}^{<}:=Fun(\mathcal{A}, Set)$ the category of copresheaves on $\mathcal{A}$, let $\mathcal{A}^{>}:=(\mathcal{A}^{op})^{<}$ the category of presheaves on $\mathcal{A}$.

Let $coCont.Fun(\mathcal{A}, \mathcal{B})$ the category of colimit preserving functors (and natural transformations).

Given the categories $\mathcal{A},\ \mathcal{B}$, (small for simplicity) a profunctor $P: \mathcal{A}\dashrightarrow \mathcal{B}$ is (defined as) as an object of $(\mathcal{A}^{op}\times$$\mathcal{B})^{<}$.

We have the isomorphisms:

$(\mathcal{A}^{op}\times \mathcal{B})^<\cong (Fun({A}^{op}, \mathcal{B}^<)\cong (coCont.Fun({A}^<, \mathcal{B}^<)$

where the first isomorphism is the elementary trasposte, the second one is the left Kan extension by the yoneda contravariant $h^-: \mathcal{A}^{op}\to $$\mathcal{A}^{<}$. Then we can view the profunctor $P$ as a (cocontinuous) functor $\widetilde{P}: {A}^{<}\to \mathcal{B}^{<}$, and given another profunctor $Q: \mathcal{B}\dashrightarrow \mathcal{C}$ the composition $Q\otimes P$ (I use the left convenction) correspond to the functor composition $\widetilde{Q}\circ $$ \widetilde{P}$ . Quite similarly we can argue about enriched categories (on a fixed monoidal symmetric (closed) one).

Now what happen about internal categories as a topos $\mathcal{E}$?

Let $Cat(\mathcal{E})$ the (2-)category of internal categories of a topos $\mathcal{E}$.

For $A\in Cat(\mathcal{E})$ let $\mathcal{E}^A$ the category of internal copresheaves on $A$ (in the literature generally indicates the category of presheaves, but make more light our notations here).

An (internal) profunctor $P: A\dashrightarrow B$ is (defined as) an object $P$ of $\mathcal{E}^{A^{op}\times B}$, this is equivalent to $(\mathcal{E}^B)^{B^\ast(A^{op})}$ where $B^{\ast}: \mathcal{E}^B\to \mathcal{B}$ canonical, and this generalizes the first isomorphism above (related to small categories on $Set$ ) to the internal categories, but what about the second isomorphism? .

I wish to know: could the category $\mathcal{E}^{A^{op}\times B}$ be equivalent (in a natural way) to a category of functors (in some sense cocontinuous) of type: $P^*: \mathcal{E}^A\to \mathcal{E}^B$ ?

Answer by Buschi Sergio for Free symmetric monoidal category on a monoidal category

2013-01-21T21:44:49Z

I hope to have done well:

Let $\mathcal{C}$ a monoidal category. Define a category $ \mathcal{C}_S$ with objets the class defined for as follow: any object of $\mathcal{C}$ is also a object of $\mathcal{C}_S$ (we call it elementary), give $X, Y \in \mathcal{C}$ then $X\cdot Y$ belong to $\mathcal{C}_S$ (we call it composite object). And for induction: If $X $ is elementary objects and $A, B$ are composite objects then $X\cdot(A)$, $ (A)\cdot X$, $ (A)\cdot (B)$ are composite objets. These are all the objets of $\mathcal{C}_S$. In few word these object are finite string of objects of $\mathcal{C}$ with a (well defined) disposition of parenthesis, on these objects is well defined the length function (the length of the underling string) by $|X|:=1$ if $X$ is elementary and with $|A\cdot B|:= |A| +|B|$, in similar way we have the function "underling string" defined as $stg(X)= X$ if $X$ is elementary and with $ stg(A\cdot B):= stg(A) \ast stg(B)$ (concatenation of string), for example if $A= (X\cdot Y)\cdot Z$ then $stg(A)=(X, Y, Z)$ and $|A|=3$. We have a map $[-]$ from the objet of $\mathcal{C}_S$ to the object of $\mathcal{C}$ that is the identity on elementary objects and with $[A\cdot B]:= [A]\otimes [B]$. The morphisms of $\mathcal{C}_S$ are of type $f: A\to B$ where is (identified to) a morphism $f: [A] \to [B]$ in $\mathcal{C}$, with identities and compositions of $\mathcal{C}$.In this way the monoidal structure $\mathcal{C}$ is translate to a monoidal structure of $\mathcal{C}_S$, and write $A \otimes B:=A\cdot B= [A]\otimes[B]$ for $A, B\in \mathcal{C}_S$.

Then we add to $\mathcal{C}_S$ the permutations (iso)morphisms $(A, \sigma): A^\sigma\to A$ where if $stg(A)=(A_1,\ldots, A_n)$ then $stg(A^\sigma)=(A_{\sigma(1)},\ldots, A_{\sigma(n)})$ and $A^\sigma$ as the some disposition of parenthesis of $A$, with composition $(A, \sigma)\circ (A^\sigma, \tau)=(A, \sigma\circ\tau)$.

And the composition of a morphisms of $\mathcal{C}_S $ with a permutation is the formal concatenation, then we get the category:

$\mathcal{C}_{sy}$ where morphisms are coherent (domain-codomain correspondence) sequences of morphism of $\mathcal{C}_S$ and permutations, the composition is by concatenation where we compose all successive morphisms of $\mathcal{C}_S$ or successive permutations everywhere is possible.

On $\mathcal{C}_{sy}$

considering the congruence $f\otimes 1_{B^\sigma}\circ 1_{A'}\otimes \sim f\otimes1_{B}\circ 1_A\otimes \sigma: A\otimes B^\sigma\to A'\otimes B$ for $f: A\to A'$ in $\mathcal{C}$ and $\sigma: B^\sigma\to B$ permutation and where $1_A\otimes \sigma: A\otimes B^\sigma\to A\otimes B$ is the permutation that in detail is $(A_1,\ldots A_n, B_1,\ldots B_m)\to (A_1,\ldots A_n, B_{\sigma(1)},\ldots B_{\sigma(m)})$ analogously for $1_{A'}\otimes \sigma$ or more un general for two permutations $\tau\otimes\sigma$. In this way we get the category $\mathcal{C}_{Sym}$.

In $\mathcal{C}_{sym}$ we define tensor product of morphisms: given $f_n\circ \sigma_n\ldots f_1\circ \sigma_1$ and $g_n\circ \tau_n\ldots g_1\circ \tau_1$ their tensorial product is $g_n\otimes f_n\circ \tau_n\otimes \sigma_n\ldots g_1\otimes f_1\circ \tau_1\otimes \sigma_1$, if these have different length we can insert identities for have morphism like above, this operation is well defined for the congruence above (and for the bifunctorial property of $\otimes$ in $\mathcal{C}$). The symmetry on $\mathcal{C}_{sym}$ is the obvious permutation $\sigma: A\otimes B \cong B\otimes A$ in detail $(A_1,\ldots A_n, B_1,\ldots B_m)\to (B_1,\ldots B_m, A_1,\ldots A_n)$ . It seems that symmetric monoidal axioms work well .

We have the (strict monoidal) functor $J: \mathcal{C}\to \mathcal{C}_{sym}$ the maps any element in its singleton, that as the universal request property for strict (resp. strong, lax) monoidal functors in symmetrical monoidal categories.

Answer by Buschi Sergio for Free monad or monad defined from an adjunction.

2013-01-13T12:31:07Z

I put this idea:

a monad in a (small) category $\mathcal{C}$ is equivalent to a 2-funtor $Mnd\to Cat$ form the free monad $Mnd$ to the 2-category $Cat$. A concrete description of Mnd is: it has one object $\ast$, the hom-category Mnd(0, 0) is the category $\Delta$ of finite ordinals and order-preserving functions, and composition $\Delta \times \Delta\to \Delta$ is ordinal sum. We have the 2-subcategory $L\subset \Delta$ ($L$ for loop) with the unique object $\ast$, full as subcategory, locally discrete.THis inclusion induce the forgetful functor $U: (T, \mu, \eta)\mapsto T$. Now the Left-Kan-extention (as Cat-enriched functor i.e. 2-functor) give the left adjoint of $U$ or in poor words the free-monad of a endofunctor.

About symmetry, braids, and pseudo-functors.

2012-12-08T18:02:41Z

Let $(\mathcal{C}, \otimes, I)$ a monoidal category, and let $\mathbb{B}(\mathcal{C})$ the bicategory (with only one object and $(\mathcal{C}, \otimes, I)$ as (monoidal) category of morphisms and cells). Let $\mathbb{B}(\mathcal{C})^{op}$ the dual respect morphisms (i.e. twisting $\otimes$, and inverting canonical isomorphisms, in the monoidal category $\mathcal{C}$). A pseudo-functor (see textarealink texttextarea) $(Q, \theta): \mathbb{B}(\mathcal{C})\to \mathbb{B}(\mathcal{C})^{op}$ that fix objects, morphisms, and cells, and is unitary (i.e. the canonical isomorphisms $Q(I)\cong I$ is the identity) is a family of isomorphism $q_{A, B}^{-1}: B\otimes A= Q(B)\otimes Q(A)\cong Q(A\otimes B)= A\otimes B $ such that:

U1) $q_{A, I}\circ r_A= l_A$

U2) $q_{I, A}\circ l_A= r_A$

where $r_A: A \cong A\otimes I$, $l_A: A \cong I\otimes A$ canonically.

and with:

$a\circ q_{C\otimes B, A}^{-1}\circ A\otimes q_{C, B}^{-1}= q_{C, B\otimes A}^{-1}\circ q_{B, A}^{-1}\otimes C\circ a^{-1}: A\otimes (B\otimes C) \to C\otimes (B\otimes A) $

or equivalently:

$A\otimes q_{C, B}\circ q_{C\otimes B, A}= a\circ q_{B, A}\otimes C\circ q_{C, B\otimes A}\circ a : (C\otimes B)\otimes A) \to A\otimes (B\otimes C) $

If we further require that $Q^{op}\circ Q=1_{\mathbb{B}(\mathcal{C})}$ then: $q_{A,B}\circ q_{B, A}=1$.

I ask:

1) is a such pseudofunctor $Q$ a symmetry?

2) What about a generalization to braids (i.e. how describe a braid as a pseudo-functor)?

Note that exist a coherence theorem: any diagram of canonical isomorphisms (from monoidal structure or from pseudofunctor data) commute

Linear (in)dependence of minors of a matrix

2012-12-16T10:04:25Z

From (Italian, very nice book):"Lezioni di Geometria Analitica e Proiettiva" by Beltrametti, CArletti, Gallarati, Bragadin (pag. 21):

Let $K$ a field, $V:= K^{n+1}$ and let $e_1,\ldots, e_{n+1}$ a base (canonical or not) of $V$. Let $W\subset K^{n+1}$ a $K$-vectorial subspace with dimention $r+1$, and let $v_1,\ldots, v_{r+1}$ a base of $W$, with

$v_m= a^1_m\cdot e_1 + \ldots a^{n+1}_m\cdot e_n$ for $1\leq m\leq r+1$

Let $M$ the matrix with ($n+1$) row's:

$x_1, a^1_1\ldots, a^1_{r+1} $

$x_2, a^2_1\ldots, a^2_{r+1} $

$\ldots, \ldots, \ldots$

$x_{n+1}, a^{n+1}_1, \ldots a^{n+1}$

(the last element is $a^{n+1}_{r+1}$)

The book assert (mentioning Kronecker theorem) that

the $r+2$-minor's of $M$ (these are $\binom{n+1}{r+2}$)

considered as linear forms (grade 1 homogeneous polynomial) on variables $x_1,\ldots, x_n$

are linearly dependent, and there are $n-r$ (and no more) linearly independent $r+2$-minors.

Is this true?

How to prove this?

Answer by Buschi Sergio for Does the dual of an object with trivial symmetry also have trivial symmetry?

2012-12-07T15:58:52Z

Brandenburg, I think that the answere is yes:

From the theory of adjunctions given $(F_k, G_k, \epsilon_k, \eta_k): \mathcal{C}\to \mathcal{C}$ for $k=1, 2$ (Maclane CWM notations), and given a natural morphism $\phi: F_2\circ F_1 \to F_1\circ F_2$ there exist a natural morphisms $\widetilde{\phi}: G_1\circ G_2 \to G_2\circ G_1$ defined as :

$G_1G_2\xrightarrow{\eta_2 G_1G_2} G_2F_2G_1G_2 \xrightarrow{G_2\eta_1 F_2 G_1G_2} G_2G_1F_1F_2G_1G_2$

$\xrightarrow{GG\phi F_2 G_1G_2} G_2G_1F_2F_1G_1G_2 \xrightarrow{GGF\epsilon_1 G} G_2G_1F_2G_2\xrightarrow{GG\epsilon_2} G_2G_1$

Considering the case $(F_1, G_1, \epsilon_1, \eta_1)= (F_2, G_2, \epsilon_2, \eta_2)$ and indicate it as
$(F, G, \epsilon, \eta)$.

By naturality, we have $GF\epsilon\ast \eta FG= \eta\ast \epsilon $, then $GGF\epsilon\ast G\eta FG= G\eta\ast G\epsilon $, then $GGF\epsilon G\ast G\eta FGG= G\eta G\ast G\epsilon G $.

Let $\phi=1$, then $\widetilde{\phi}= GG\epsilon\ast GGF\epsilon G\ast G\eta FGG\ast \eta GG = GG\epsilon\ast G\eta G\ast G\epsilon G \ast \eta GG =$

$=G(G\epsilon\ast \eta G)\ast (G\epsilon \ast \eta G)G =1_G\ast1_G=1_G $.

Now we use this proof for a 2-category with a only one object, (essentially a strict monoidal category), and then to a bicategory with one object (essentially a monoidal category).

About the closed structure on the modules of a monoidal closed symmetrical category

2012-11-25T21:59:17Z

Let $(\mathscr{C}, \otimes , I)$ monoidal category, a monoid $(R, e_R, \mu )$ is a object $R \in \mathscr{C}$ with morphisms $e_R: I \to R$, $\mu: R \otimes R \to R$ with the well knowed unital e associative commutative diagram.

If $(\mathscr{C}, \otimes , I)$ has a symmetry (is enought a braid) then the tensor prodoct $M \otimes N $ of two objects, each one with a monoid structure, has a natural monoid structure, then we have a monoidal symmetrical (braided) category of monoid of $(\mathscr{C}, \otimes , I)$. Given a monoid $R$ a left $R$-module $(M, \mu)$ is a object $m \in C$ with a morphism $\mu: R \otimes M \to M $ with usual unitary and associative diagram, similarly are defined the right modules.

Now we suppose that $(\mathscr{C}, \otimes, [-, ?], I)$ ia a monoidal closed, symmetrical category, given a right $R$-module $M$and a a left $R$-module $N$, In analogy to the classical algebra context, let $[M, N]^R $ the kernel of the two morphisms gived by the compositions:

$[M, N] \xrightarrow{- \otimes R} [M \otimes R, N \otimes R ] \xrightarrow{[1, \cong ]} [M \otimes R , R \otimes N ] \xrightarrow{[1, \mu_N]}[R \otimes M, N ]$

$[M, N] \xrightarrow{ [\mu_M, 1] } [R \otimes M, N ]$

How to prove that $[M, N]^R$ is a left $R$-module (for the structure inducted by $M$, and inducted by $N$) ?

This is mentioned on §3 of the Stefan Schwede & Brooke Shipley article 'Algebras and modules in monoidal model categories'

http://arxiv.org/abs/math/9801082

I know that A. Kock studied the monoidal -closed structure of algebras of a (commutative) triple in monoidal closed symmetrical categories (then a much more general and complex result).

I ask where I can find a explicit but rigorous proof.

Answer by Buschi Sergio for When do Kan extensions preserve limits/colimits?

2012-11-24T19:12:20Z

Let $(a_i: A_i\to A)_{i\in I}$ with $I\in Cat$ a universal cocone in a category $\mathcal{A}$, and let $H: \mathcal{B}\to \mathcal{A}$.

We ask when:

for any $F: \mathcal{B}\to \mathcal{C}$ such that:

exist $L:=Lan_H F$ punctually (or at least it exist for the objects $A,\ A_i$ of the above diagram i.e. exists $L(A_i):=\varinjlim_{(B, b)\in H\downarrow A_i} F(B)$ and $L(A):=\varinjlim_{(B, b)\in H\downarrow A} F(B)$).

we have that $L(A)=\varinjlim_i L(A_i)$.

Consider the colimit category $\widehat{HA}:=\varinjlim_i H\downarrow A_i$, the functors $F\circ \pi_{A_i}:H\downarrow A_i\to \mathcal{B}\to \mathcal{C}$ induce a funtor $\hat{F}: \varinjlim_i H\downarrow A_i \to \mathcal{B}$ and is not hard verify that $\varinjlim_i L(A_i)=\varinjlim_i \varinjlim_{(B, b)\in H\downarrow A_i} F(B)= \varinjlim_{(B, b)\in \widehat{HA}} F(B)= \varinjlim \widehat{F}$.

Then the natural morphisms $\phi: \varinjlim_i L(A_i)\to L(A)$ is induced by the natural functor $\Phi: \widehat{HA}\to H\downarrow A $, then $\phi$ is a isomorphism (for any $F$ such that..) iff the functor $\Phi$ is final i.e. iff each morphism $H(B)\to A $ has a factorization on some $H(B')\to A_i\to A$ (through a morphism $B\to B'$) and two such factorization are connected in $H\downarrow A$.

Answer by Buschi Sergio for Cocomplete but not complete abelian category

2012-11-22T20:37:21Z

I'm no totally sure (as ever), If no I hope could suggest some ideas..

Let $fAb$ the abelian category of finite abelian groups, and let $\mathcal{C}:= Ind(fAb)$ its ind-category, this is the full category of presheaves on $fAb$ isomorphic to a filtred diagram of representable. Now form usual literature (e.g. Artin MAzur "Etale Homotopy" appendix) $\mathcal{C}$ is abelian, (then has finite sums), and has filtered colimits then has (small) sums, then is cocomplete.

Consider a countable numeration of finite cyclic groups $(C_n)_{n\in \mathbb{N}}$, and suppose that exist the product $P:= \prod_n h_{C_n}$ in $\mathcal{C}$ of associate representable of the $C_n$'s, let $P\cong \varinjlim_{i\in I} h_{G_i}$ for some direct diagrams of finite abelian groups $G_i$. We have a split monomorphisms $\delta: \sum_n h_{C_n}\to P$, i claim the the family of maps $h_{C_n}\to \sum_n h_{C_n}\to P$ is epimorphic, this follow because the the family $G_i\to P$ is epimorphic, and any $G_j$ is a (finite) sums of cyclic groups. But then $\delta$ is a epimorphism, then a isomorphism. Now fix a cyclic group $C_{m}\neq 0$, and consider $(h_{C_m}, \sum_n h_{C_n})\cong \bigoplus_n fAb(C_m, C_n)$ (the sum is a direct colimits of finite sums, and finite sums are representable by a biproduct) and each elemts of this sum as all $0$'s but finite components, but this isnt true for $(h_{C_m}, P)\cong \prod_n fAb(C_m, C_n)$, and considering that $1_{C_n}:= \pi_i\circ \delta\circ \epsilon_i : h_{C_n}\to \sum_n h_{C_n}\to P \to h_{C_n} $ we get a absurd condition.

A (too easy) normalization of a lax-funtor between 2-categories ?

2012-11-01T07:59:08Z

Let $(F, \phi): \mathscr{A}\to \mathscr{B}$ a lax-functor between 2-categories. In the setting of 2-categories the axiom of lax-functor become:

Ul) $1: F(f)=F(f)\circ 1_{F(X)}\xrightarrow{1\circ \phi_X}F(f)\circ F(1_X)\xrightarrow{\phi_{f, 1}} F(f\circ 1_X)= F(f)$.

For $f: X \to Y$ in $\mathscr{A}$.

Ur) $1: F(f)= 1_{F(Y)}\circ F(f)\xrightarrow{\phi_X\circ 1} F(1_Y)\circ F(f)\xrightarrow{\phi_{1, f}} F( 1_Y \circ f)= F(f)$.

For $f: X \to Y$ in $\mathscr{A}$.

UA) The compositions $F(h)\circ F(g)\circ F(f) \xrightarrow{\phi_{h, g}\circ 1} F(h\circ g)\circ F(f) \xrightarrow{\phi_{h\circ g, f}\circ 1} F(h\circ g\circ f)$ and

$F(h)\circ F(g)\circ F(f) \xrightarrow{1\circ \phi_{g, f}} F(h)\circ F(g\circ f) \xrightarrow{\phi_{h, g\circ f}} F(h\circ g\circ f)$ are equal.

For componibile morphisms $h, g, f$.

I call $(F, \phi)$ normal (or unitary) if in the axioms $(Ul)$ and $(Ur)$ above all arrows are identities.
Gived $(F, \phi)$ (general) I define a normal lax.functor $(\tilde{F}, \tilde{\phi})$ that is the some of $F$ on objects and on non-identity morphisms, with of course $\tilde{F}(1_X)=1_{F(X)}$, $\widetilde{\phi}_{X}= 1: \tilde{F}(1_X)\to 1_{F(X)} $, and with $\tilde{\phi}_{g, f}$ defined as:

$ \phi_{g, f}: F(g)\circ F(f)\to F(g\circ f)$ if $f$ and $g$ arent identity, and the obvious identity if $f$ or $g$ is a identity. I checked (easly) that axiom $(UA)$ is true for $(\tilde{F}, \tilde{\phi})$, then $(\tilde{F}, \tilde{\phi})$ is a lax.funtor.

I ask if this (very easy normalization) is just know in literature ( I dont know), and if is right (I'm sure its right, but I'm no too sure of myself).

Edit: If in $(Ul)$ all arrows are identies, this imply the some in $(Ur)$ (i.e. $(F, \phi)$ is normal)?

My motivations is a generalization of the J.W. Gray concept of quasi-functors to lax.functors (Gray gived this definition for 2-functors).

EDit As Jonathan Chiche observed, these is the obvious condition than a lax.functor induce hom-functors between hom-categories (I am ashamed for this rough oversight). Anyway if the canonical morphism $\phi_X: 1_{FX}\Rightarrow F(1_X)$ is a isomorphism then the mine definition preserving this funtorialiy, only need a easy correction based on the following observation: the cells of type $\sigma: 1_{FX}\Rightarrow W$ are in bijections with the cells of type $\sigma': F(1_X)\Rightarrow W$ and the cells like $\tau: W\Rightarrow 1_{FX}$ are in bijections with the cells of type $\tau': W\Rightarrow F(1_X)$. With this correction I seems that also the naturality of $\tilde{\phi}$ work well.

Answer by Buschi Sergio for Where is it rigorously stated and proved that the definition of lax functor implies that the generalized cocycle condition holds for an arbitrary number of composable $1$-cells?

2012-11-02T20:04:27Z

For composable couple of morphisms $g\circ f$ let

$T(g, f): F(g)\circ F(f) \Rightarrow F(g\circ f)$ the canonical cell.

For a triple of composable $h\circ g\circ f$ let $T_r(h, g, f):=T(T(h, g),f)$ (A short way to write $T(hg, f)\ast T(h, g)f$)

and $T_l(h,g,f):= T(h,T(g, f))$ for the axiom of coherence $T'_3=T''_3$.

If we have composable $n$ morphisms $f_n\circ \ldots\circ f_1$ for the various (associativity) coherent disposition of parenthesis (as in a no necessarily associative binary composition) we have a cell $F(f_n)\circ \ldots F(f_1) \to F(f_n\circ \ldots f_1)$, we want to prove that all these cells are equal, for $n=3$ this is true. For induction we suppose the assert for 3, ..., n and indicate with $T^k$ ($3\leq n\leq n$) the unique composition cell $F(f_k)\circ \ldots F(f_1) \to F(f_k\circ \ldots f_1)$. Let $T_{n+1}: F(f_{n+1})\circ \ldots F(f_1)\xrightarrow{1\circ T^n} F(f_{n+1})\circ F(f_n\circ \ldots f_1)\xrightarrow{T} F(f_{n+1}\circ f_n\circ \ldots f_1) $, let $T': F(f_{n+1})\circ \ldots F(f_1)\to F(f_{n+1}\circ f_n\circ \ldots f_1) $ associated to some coherent parenthesis disposition. Now if $f_{n+1}$ is not inside a parenthesis (do not allow the total parenthesis on the whole string) we have that $T'(f_{n+1}, \ldots f_1): F(f_{n+1})\circ F(f_n)\circ \ldots F(f_1)\xrightarrow{1\circ T} F(f_{n+1})\circ F(f_n\circ \ldots f_1)$ and this is $T_{n+1}$, if $f_{n+1}$ is inside a parenthesis consider the maximal parenthesis containing $f_{n+1}$ and let $f_{n+1},\ldots f_{i+1}$ the part of string contained in this parenthesis, the case $i=n-1$ follow as above,

if $i< n-1$ then

$T'(f_{n+1}, \ldots f_1)= T(T^{n-i}(f_{n+1},\ldots f_i), T^{i}(f_i,\ldots f_1))=$

$T(T((f_{n+1},T^{n-i-1}(f_n, \ldots f_i)), T^{i}(f_i,\ldots f_1))=$

$T((f_{n+1},T^n(f_n, \ldots f_1))=$

$T_{n+1}(f_{n+1}\circ \ldots f_1) $.

This is only a traslation from the classical algebra theorem.

What about lax-coherence results?

2012-11-02T15:25:15Z

As well know in category theory (see Mac Lane book) there are (very useful, and theoretically deep) coherent theorem for various categorical setting: monoidal, symmetric monoidal, braided monoidal (tortile..) bicategory, and pseudofunctor (coherence of canonical morphisms of a pseudofunctors and these of its codomain bicategory, a further case about is ifrations by clivage...).

What about the (partial of course) coherence criterion of a lax.functor between bicategories (lax. generalization of the pseudo.functor result )?

Answer by Buschi Sergio for Monoidal structure on a category with products and with terminal object

2012-11-02T07:26:39Z

Is very easy prove that $(Set, \times, 1)$ is monoidal (by elements checking). Now let $\mathcal{C}$ a category by finite product $\times$ and (then) with a final object $1$. Consider the axioms of monoidal category for $(\mathcal{C}, \times , 1)$ stated by diagrams (see for example p.462 of "Closed Categories" by Eilenberg & Kelly, LA Jolla 1967), now it remains to prove that these diagrams are commutative. COnsider a such diagram $\textbf{D}$ and a (general) object $X\in \mathcal{C}$ and the representable $(X, -): \mathcal{C}\to Set: A \mapsto (X, A)$, acting by $(X, -)$ on this diagram, we get a similar diagram in $Set$, say $X(\textbf{D})$, and $(X, -)$ preserve the product $\times$ and the final object $1$, now we just know that $(Set, \times, 1)$ is monoidal, then $X(\textbf{D})$ is commutative. Because this is true for each object $X$, by Yoneda lemma follow that $\textbf{D}$ is commutative (more easily observe that given $f, g: A \to B$, if $(X, f)=(X, g): (X, A)\to (X, B) $ for each $X$ then $f=g$ (consider $X=A$ and $1_A$)).

A Question on Auto-Adjoint functors

2012-09-02T21:18:41Z

Let $F: \mathscr{C}\to \mathscr{C}^{op}$, with an adjoint $G$, and $\eta: 1_\mathscr{C} \Rightarrow G\circ F $ and $\varepsilon: F\circ G\Rightarrow 1_{\mathscr{C}^{op}}$ with components (in $\mathscr{C}$):

$\eta_M: M\to G(F(M))$ and $\varepsilon^{op}_N: N\to F(G(N))$ such that the compositions (in $\mathscr{C}$):

$F(M) \xrightarrow{\varepsilon^{op}_{F(M)}} FGF(M) \xrightarrow{F^{op}(\eta_M)}F(M)$ and

$G(N) \xrightarrow{\eta_{G(N)}}GFG(N) \xrightarrow{G(\varepsilon_N)}G(N)\ $ are units.

Moreover suppose $G=F^{op}$ and $\eta$ a isomorphis transformation. Follow that $F$ is full, faithfull and reflect isomorphisms (consider $F(f)\circ \eta_M=\eta_{M'}\circ f$ for $f: M\to M'$), then also $G=F^{op}$ reflect isomorphisms, from above: $\varepsilon$ is Iso. Observe that the components of $\eta: 1_\mathscr{C} \Rightarrow F^{op}\circ F $ and $\varepsilon^{op}: 1_{\mathscr{C}}\Rightarrow F^{op}\circ F$ are of type: $\eta_M: M\to F\circ F(M)$ and $\varepsilon^{op}_M: M\to F\circ F(M)$.

Question: Is true that $\eta =\varepsilon^{op}$ ?

On Benabou and Johnstone definition of "locally small" fibred (indexed) category

2012-08-19T10:04:26Z

Let $P: \mathcal{C}\to\mathcal{S}$ a fibration ($\mathcal{S}$ with finite limits).

In "Sketches of an Elephant I", pag. 272 P. Johnstone define $P$ a locally small if:

given any two objects $X, Y\in \mathcal{C}$ (let $A=P(X),\ B=P(Y)$) there exist a arrow $(a, b): I\to A\times B$ and a morphisms $f: a^\ast X\to b^\ast Y$ in the fibre $\mathcal{C}(I)$ such that given any $(c, d): J\to A\times B$ and any $g: c^\ast X\to d^\ast Y $ in $\mathcal{C}(J)$ there exists a unique $u: J\to I$ such that $a\circ u=c,\ b\circ u=d$ and $u^\ast(f)\ \dot{=}\ g$ (where " $\dot{=}$ " means up to canonical isomorphisms).

In LNM 661 "Indexed Categories and its Applications" p. 40, the authors post the J. Benabou definition: $P$ is a locally small if:

for any $I\in\mathcal{S}$ and every $X, Y\in \mathcal{C}(I)$ there exist a morfisms ${}^Ih_{A, B}: {}^IH_{A, B} \to I$ such that for any $\alpha: J\to I$ there is a bijection between the morphisms:

$\alpha \to {}^Ih_{A, B}$ in $\mathcal{S}\downarrow I$

and the morphisms $\alpha^\ast(X)\to \alpha^\ast(Y)$ in $\mathcal{C}(I)$

I ask: how are related (if they are) these two definitions?

Edit:

Johnston define "locally small" as the (what he define) comprehension scheme for the inclusion $2\to \underline{2}$

($2$ is the discrte category ${0, 1}$ and $\underline{2}$ is $0\to 1$ plus identities).

THis means that the composition funtor $Rect(\underline{2},\ \mathcal{C} )\to Rect(2,\ \mathcal{C} )$ has a right adjoint, where $Rect(\mathcal{D},\ \mathcal{C} )$ is the category with objects the diagrams $d: \mathcal{D}\to \mathcal{C}$ with vertical edges (i.e. mapped to identities by $P$), and with morphisms the transformations with all components cartesians. (articulating this, get the definition given at the beginning).

THe BEnabou definition is equivalent to the more strict assert:

the composition funtor $Rect_P(\underline{2},\ \mathcal{C} )\to Rect_P(2,\ \mathcal{C} )$ has a right adjoint, where $Rect_P(\mathcal{D},\ \mathcal{C} )\subset Rect(\mathcal{D},\ \mathcal{C} )$ is given by diagram $d: \mathcal{D}\to \mathcal{C}$ with $P\circ d$ constant (i.e. maps all on some object $A$ and its identity $1_A$) and with morphisms the transformations with all components cartesians and mapped by $P$ on the same morphism.

THen the Johnstone definition imply the BEnabou one: by restriction of adjunction, observing that $2$ is just the objects of $\underline{2}$ then the condiction on transformations components is preserved.

From the CHuck answere, the reverse i true too:

LEt $X, Y\in \mathcal{C}$ and let $A=P(X),\ B=P(Y)$, considering $\pi_A: A\times B\to A,\ \pi_B : A\times B\to B$, put $I:=A\times B$ and considering $\pi_A^\ast(X),\ \pi_B^\ast(Y)$ on the $I$-fibre, appling the BEnabou condiction follow the Johnstone one.

Then the initial request has the follow generalization:

given a functor $F: \mathcal{D'}\to \mathcal{D} $

Is true that:

If the natural funtor

$Rect(\mathcal{D},\ \mathcal{C} )\to Rect(\mathcal{D'},\ \mathcal{C} )$ has a right adjoint

then

$Rect_P(\mathcal{D},\ \mathcal{C} )\to Rect_P(\mathcal{D'},\ \mathcal{C} )$ has a right adjoint ?

(from above, I seem the if $F$ is surjective on objets then the proposition is true)

Is true the reverse?, with such conditions on $F$ the reverse can be true?

Answer by Buschi Sergio for functors unique up to self-equivalence of the source category

2012-08-17T19:06:54Z

This is a mine observation (from old personal paper), may be can help. I hope my English is understandable

I thought about this question in correlation to proposition 2.2.1 p.31 of "Indexed categories and its Applications" LNM 661:

Give a functor $F:\mathcal{C}^{op}\to Cat$ (i.e. a strict indexed category) if its objets and morphisms presheaf are representable: $\mathcal{A}(C_0, -)\cong F_0$, $\mathcal{A}(C_1, -)\cong F_1$ by Yoneda lemma domain, codomain the transformations $d_0: F_1\Rightarrow F_0$, $d_1: F_1\Rightarrow F_0$ ecc. describe a internal category $\underline{C}$ of $\mathcal{A}$ and a isomorphism $F\cong [\mathcal{C} ]$ where $[\mathcal{C}]$ is the presheaf of (small) categories defined as $|\mathcal{C}|_0=\mathcal{A}(A, C_k)\ k=0, 1$ ecc.

But what if $F$ is a pseudo.functor? ($F_0$, $F_1$ aren't presheaves, for the composition we haven't the strict coherence, but accompanied by automorphisms on codomain..) The authors of the reference above consider $F_0(A)$ as the class of (canonically) isomorphic object of $|F(A)|_0$, and for $F_1(A)$ as the class of (canonically) isomorphic morphisms (as objects of $F(A)^{\underline{2}}$). At first reading I seem that this reduction to "pseudo-skeletons" avoid the trouble of the canonical isomorphisms. I remember that the skeleton of a (small) category is make by a chose of a object for any isomorphic call, and make the full subcategory. But how is related a small category $\mathcal{C}$ with its pseudo-skeleton $pS(\mathcal{C})$?

Of course we have the natural functor (make the quotient maps) $\pi: \mathcal{C}\to pS(\mathcal{C})$, now, try to define a reciprocal (equivalence we hope) functor $P$, merely a "chose functor": then $P$ map any object $\xi\in pS(\mathcal{C})$ to a its representant $X_{\xi}\in \mathcal{C}$ with $[X_\xi]=\xi$, and any morphism $\alpha: \xi\to \xi'$ to a its representant $f_\alpha: X_\xi\to X_{\xi'}$ with $[f_\alpha]=\alpha$,

(observe that a different chose $g_{\xi}$ is related to $f_\xi$ by automorphisms on domain and codomain $X_\xi$, $X_{\xi'}$ this is a generalization to the initial question)

we can chose $P(1_\xi)=1_{X_\xi}$ of course, but in general $f_{\gamma\circ\xi}\neq f_{\gamma}\circ f_\xi$ but these are related by automorphisms on domain and codomain.

Then I consider 2-category $G(\mathcal{C})$ make as follow: take the usual skeleton $Sk(\mathcal{C})$, then for $f, g: X\to Y$ define 2-cell $f\Rightarrow g$ as a couple of isomorhisms $s: X\to X,\ t: Y\to Y$ with $t\circ f=g\circ s$ (with natural composition and identities) then I have a correct 2-equivalences (make by a 2-functor and a pseudo-functor)) from $G(\mathcal{C})$ and its local skeleton (take skeleton on Hom-categories) that is just the pseudo-skeleton $pS(\mathcal{C})$ .

Answer by Buschi Sergio for Are bicategories of lax functors also bicategories of of pseudofunctors?

2012-08-15T13:19:01Z

For 2-categories there is a proof on "FOrmal CAtegory Theory I" of J. W. Gray LNM 391 (see I,4.23 pag. 92) . There is cited that J. Benabou did a general proof for bicategories.

Does not exists lax.limits of lax.functors in CAT?

2012-08-13T19:45:05Z

Premise

1) Give a lax.functor: $\textbf{C}: \mathcal{A}^{op}\to CAT $ a op.lax.cocone (resp. lax.cocone) $\check{\phi}: \textbf{C}\Rightarrow \mathcal{C}$ form $\textbf{C}$ to a category $\mathcal{C}$ is a op.lax.transformation (resp. lax.transformation) form $\textbf{C}$ to the constant lax.functor $k_\mathcal{C}$ (that send all on $\mathcal{C},\ 1_{\mathcal{C}},\ 1_{1_{\mathcal{C}}}$) (see link text).

2) Give a prefibration (see [S], p.132) $P: \mathcal{C}\to \mathcal{A}$ with a preclivage, we can associate to it the lax.functor:

$\textbf{C}: \mathcal{A}^{op}\to CAT $ with

$\textbf{C}(A)=\mathcal{C}(A)$ (the fibre of $P$ on $A$)

$\textbf{C}(f)=f^\ast: \textbf{C}(B)\to \textbf{C}(B)$ for $f: A\to B$ (the inverse image by $f$ )

with $C_{g, f}=c_{c, f}: \textbf{C}(g)\circ \textbf{C}(f)\Rightarrow\textbf{C}(g\circ f)\ g\circ f: A\to B\to C$

(where $c_{g, f}$ come from the clivage).

There exist a natural op.lax.cocone $\check{\phi}: \textbf{C}\to \mathcal{C}$ gived by

$\check{\phi}_A: \textbf{C}(A)\to \mathcal{C}$ (natural inclusion), $\check{\phi}_f: \check{\phi}_A\circ \textbf{C}(f)\Rightarrow \check{\phi}_B$ with $\check{\phi}_A(Y)=\theta_f: f^\ast(X)\to Y$ (canonical morphism of inverse image of the clivage).

And this op.lax.cocone is universal, i.e. it generalize the colimits concept for op.laxcocone (see [G] p.201), in other word $\mathcal{C}$ is a op.lax.colimit of $\textbf{C}$.

I ask:

Exist a lax.colimit of a lax.functors $\textbf{C}: \mathcal{A}^{op}\to CAT $ associated to a prefibration $P: \mathcal{C}\to \mathcal{A}$?

A universal lax.cocone have to be a family:

$\check{\phi}_A: \textbf{C}(A)\to \mathcal{C}$ with

$\check{\phi}_f: \check{\phi}_B \Rightarrow \check{\phi}_A\circ \textbf{C}(f)$

I tried to define $\mathcal{C'}$ with object the couple $(X, A)\ A\in \mathcal{A}, X\in \textbf{C}(A)$ and morphisms $(\phi, f): (B, Y)\to (A, X) $ with $f: A\to B$ and $\phi: f^\ast(Y)\to X$, but there is a trouble for the definition of composition (need to reverse the arrow $c_{g, f}: f^\ast\circ g^\ast\Rightarrow (g\circ f)^\ast$). Or tried to define $\mathcal{C'}$ with objects as above but reversing $\phi$, i.e. $\phi:X\to f^\ast(Y)$ (this is the Grothendieck construction for the op.prefibration $P^{op}$) but there is a trouble for defining $\check{\phi}_A: \textbf{C}(A)\to \mathcal{C'}$ (naturally: $\textbf{C}(A)\subset \mathcal{C'}^{op}$).

[S] A. Grothendieck SGA1 cap. VI (SEMINAIRE DE GEOMETRIE ALGEBRIQUE, 1960-61) \end{document}

[G] J.w. Gray "FOrmal Category Theory I" LNM N°391

A question about prefibrations and lax tranformations

2012-08-09T16:38:19Z

Let $P: \mathcal{C}\to \mathcal{A}$ be a functor. We call a morphism $v$ of $\mathcal{C}$ vertical (over $A\in \mathcal{A}$) if $P(v)=1_A$; then we have the fibre category $\mathcal{C}_A$ of vertical morphisms over $A$. We call a morphism $f: X\to Y$ in $\mathcal{C}$ cartesian if for $f': X'\to Y$ with $P(f)=P(f')$, there exists a unique vertical morphism $v: X'\to X$ such that $f'=f\circ v$.

If $f: A\to B$ is in $\mathcal{A}$ and $Y\in \mathcal{C}(B)$, we call a cartesian morphism of type $\theta_f(Y): f^\ast Y\to Y$ with $P(\theta_f)=f)$ a cartesian $f$-lifting of $Y$. If there exists such for each $f$ and $Y$, we call $P$ a prefibration, and a precleavage is a choice of $\theta_f(Y)$ for each $f$ and $Y$. If we choose $\theta_{1_A}=1_A$, the preclivage is called normal.

After fixing a cleavage, we obtain a functor $\theta_f: \mathcal{C}(B)\to \mathcal{C}(A)$, and for $X\xrightarrow{f}Y\xrightarrow{g}Z$ there is a transformation $c_{g, f}: f^\ast\circ g^\ast\Rightarrow (g\circ f)^\ast$ (considering $ \theta_g(Z)\circ \theta_f(g^\ast(Z))$). Thus, we have a lax functor $\textbf{C}: \mathcal{A}^{op}\to CAT$ defined as $\textbf{C}(A)=\mathcal{C}(A)\ A\in\mathcal{A}$, $\textbf{C}(f)=f^{\ast}$,

$C_{g,f}=c_{g, f}: \textbf{C}(f)\circ \textbf{C}(g)\Rightarrow \textbf{C}(g\circ f)$.

Let $P: \mathcal{C}\to \mathcal{A}$ and $Q: \mathcal{D}\to \mathcal{A}$ be prefibrations with fixed normal cleavages. If $T: \mathcal{C}\to \mathcal{D}$ is a functor over $\mathcal{A}$, i.e. with $Q\circ T= P$, then by restriction we have functors $T_A: \textbf{C}(A)\to \textbf{D}(A)$. Applying $T$ to $\theta_f(Y)$ and considering the cartesian $\theta_f(T(Y))$ (in $\mathcal{D}$), we have a natural vertical morphism $T_f(Y): T(f^\ast(Y))\to f^\ast(T(Y))$, which defines a transformation $T_f: T_A\circ \textbf{C}(f)\Rightarrow \textbf{D}(f)\circ T_B$. The data $(T_A, T_F)_{A, f}$ is what is called an oplax transformation. In the other direction, from an oplax transformation $(T_A, T_F)_{A, f}$ we can make a functor $T: \mathcal{C}\to \mathcal{D}$ over $\mathcal{A}$ (considering that each morphism over $\mathcal{C}$ or in $\mathcal{D}$ has a unique factorization as a vertical morphism followed by a cartesian morphism).

Thus the oplax transformations are identified with functors $T$ over $\mathcal{A}$.

My question is:

If (instead of an oplax tranformation) we consider a lax tranformation $T$, i.e. a family

$T_A: \textbf{C}(A)\to \textbf{D}(A)$ and $T_f: \textbf{D}(f)\circ T_B \Rightarrow T_A\circ \textbf{C}(f)$

with the coherence conditions:

$T_{gf}\ast (d_{g, f})\circ T_C)=$

$(T_A\circ c_{g, f})\ast(T_f\circ \textbf{C}(g))\ast(\textbf{D}(f)\circ T_g): \textbf{D}(f)\circ\textbf{D}(g)\circ T_C\Rightarrow T_A\circ \textbf{D}(f)\circ\textbf{D}(g)$

(the unitary condition being unnecessary, by our choice of normal cleavages).

Is there some way to represent $T$ as a categorical construction?

About the coherence of pseudo.colimits and limits of base categories.

2012-08-10T23:13:53Z

premises:

Let $Clv(\mathcal{A})$ the category with objects the fibrations on $P:\mathcal{C}\to \mathcal{A}$ with a fixed clivege, and with cartesian functors $T: \mathcal{C_1}\to \mathcal{C_2}$ on $ \mathcal{A}$ (i.e. $P_2\circ T=P_1$).

Let $Spl(\mathcal{A})\subset Clv(\mathcal{A})$ the subcategory of split clivages, and clivage preserving functors on $ \mathcal{A}$.

We have a equivalence of $Clv(\mathcal{A})$ with the category

$p$-$Fun(\mathcal{A}^{op})$ of pseudo-functors $\textbf{C}: \mathcal{A}^{op}\to CAT$ and pseudo-transformations

and a equivalence of $Spl(\mathcal{A})$ with the category
$2$-$Fun(\mathcal{A}^{op})$ of 2-functors $\textbf{C}: \mathcal{A}^{op}\to CAT$ and 2-transformations.

For $F: \mathcal{A}\to \mathcal{B}$ we have the pullback functor $F^\ast: Clv(\mathcal{A})\to Clv(\mathcal{B})$ and its restriction $F^\ast: Spl(\mathcal{A})\to Spl(\mathcal{B})$

this latter correspond (by the equivalences above) to the (right) composition by $F^{op}$:

$F_\ast: 2$-$Fun(\mathcal{B}^{op})\to 2$-$Fun(\mathcal{A}^{op})$

and this has a left-adjoint and a right-adjoint give by the respective Kan-extensions (chose $\mathcal{A}$ and $\mathcal{B}$ small or ample the universe) then also $F^\ast$ above as a left adjoint and a right adjoint.

Let $F_\bullet: Clv(\mathcal{B})\to Clv(\mathcal{A})$ the left adjoint

in the equivalent form $F_\bullet: Spl(\mathcal{B})\to Spl(\mathcal{A})$ it is give by:

$F_\bullet(P)(B)= \varinjlim P\circ (\pi^B_F)^{op}= \varinjlim_{a: B\to F(A)} P(A)$

with $\pi^B_F: B\downarrow F\to \mathcal{A}$ natural.

Let $Oub_F: Spl(\mathcal{B})\xrightarrow{F^\ast} Spl(\mathcal{A})\xrightarrow{U} Clv(\mathcal{A})$

($U$ is the trivial forgetful inclusion).

From [G] 2.4.2.1( p.38) we have a left adjoint (is a 2-adjoint):

$Gau_F$ of $Oub_F$, that in term of pseudo.funtors, for $P\in p$-$Fun(\mathcal{A}^{op})$ this is give by:

$Gau_F(P)(B)= \underrightarrow{LIM}\ P\circ (\pi^B_F)^{op}$

where $\underrightarrow{LIM}$ is the pseudo-colimit operator.

Now, for $F=1_\mathcal{A}$ we get a left adjoint $L$ of the inclusion $U: Spl(\mathcal{A}\to Clv(\mathcal{A})$:

$L(P)(A)= \underrightarrow{LIM}\ P\circ (\pi_A)^{op}$ with $\pi_A: A\downarrow \mathcal{A}\to \mathcal{A}$.\

But then $Gau_F\cong F_\bullet\circ L$

and then $\underrightarrow{LIM}\ P\circ \pi^B_F\ = \ \varinjlim_{(a, A)\in B\downarrow F} \underrightarrow{LIM}(P\circ \pi_A)$\

Now, in the category $CAT\downarrow \mathcal{A}$ we have that $\pi^B_F= \varinjlim_{(a, A)\in B\downarrow F} \pi_A $

or in more explicit way (in terms of the domains categories):

$B\downarrow F= \varinjlim_{a: B\to F(A)} A\downarrow \mathcal{A}$. And then $(\pi^B_F)^{op}= \varinjlim_{(a, A)\in B\downarrow F} (\pi_A)^{op}$

Then my question is:

Let $\mathcal{C}=\varinjlim_{i\in I}\mathcal{C}_i$ a colimit of categories and $P: \mathcal{C}\to CAT$ a pseudofutor, the natural coproiections $\varepsilon_i: \mathcal{C}_i\to \mathcal{C}$ induce a morphism $e_i: \underrightarrow{LIM} (P\circ \varepsilon_i)\to \underrightarrow{LIM} P $, is this a pseudo-colimit? in other therm is EVER true that:

$\underrightarrow{LIM} P=\varinjlim_{i\in I}\ \underrightarrow{LIM} P_{|\mathcal{C}_i}$ ?

What for lax.colimit?

Bibliography: [G]: J , Giraud "Cohomologie non Abelienne"

Is a colimit of fibration still a fibration?

2012-08-06T18:34:16Z

Let $\mathcal{C}$ a category.

Let $Fib(\mathcal{C})$ the category of fibrations (on $\mathcal{C}$) with morphisms the cartesian functors $T: (\mathcal{A}, P)\to (\mathcal{B}, Q)$

i.e. $T: \mathcal{A}\to \mathcal{B}$ with $Q\circ T=P$ (where $P: \mathcal{A}\to \mathcal{C}$, $P: \mathcal{B}\to \mathcal{C}$ are fibrations on $\mathcal{C}$).

From literature (es. "Categorical logic and Type THeory" B.Jacobs) the pullbak of two fibration (i.e. the product in $Fib(\mathcal{C})$) is still a fibration, and this has a easy generalization to a multi-pullback i.e. a the inclusion $Fib(\mathcal{C})\subset CAT\downarrow \mathcal{C}$ create (small) product, the some is true for the kernel of a couple, then the inclusion above create all (small) limits. THe some is easly true for coproducts.

I ask: do the inclusion $Fib(\mathcal{C})\subset CAT\downarrow \mathcal{C}$ create also cokernels? Counterexamples?.

Observation $Fib(\mathcal{C})$ is equivalent to the category of pseudo-functors $P: \mathcal{C}^{op}\to CAT$ with pseudo-transformations as morphisms, and the punctual limits or colimits of pseudofunctors is still a pseudofuntors (I seems yes, anyway this is true for funtors and natural tranformations considering the category $sFib(\mathcal{C})$ of fibrations with split clevages and clevage preserving functors ), then $Fib(\mathcal{C})$ (or at least $sFib(\mathcal{C})$) is complete and cocomplete.

Answer by Buschi Sergio for Geometric invariant theory for geometers

2012-08-03T11:02:33Z

For a "more classical" point of view:

"An introduction to Invariants and Moduli " by S. Mukai.

Fogarty J. "Invariant theory" (Benjamin, 1969)

For a introduction to Mumford's:

P. E. Newstead, "Introduction to Moduli Problems and Orbit Spaces"

Anyway you have to learn (before or after) a Gothendieck-categorical background:

Fondements de la géométrie algébrique (Grothendick). The Hilbert schema chapter is very important (need the Hartshorne "Algebraic Geometry" as base)

Or in more gentle way: Fundamental Algebraic Geometry. Grothendieck's FGA Explained - Fantechi B., Göttsche, L., Illusie L.

Answer by Buschi Sergio for Is there a categorical treatment of dynamical systems?

2012-08-03T10:38:00Z

See "J. de Vries, Topological transformation groups 1, A cat.approach" and similar works about topological group actions from a categorical perspective.

(http://oai.cwi.nl/oai/asset/12605/12605A.pdf or search for "Topological transformation groups" (any author) in http://repository.cwi.nl/).

THese is a account of it in the last article in Springer Lecture notes in MAth n. 540 (catagorical topology).

ABout homotopical question of Topological transformation groups see the Tammo Tom Dieck book (http://www.amazon.com/Transformation-Groups-Gruyter-Studies-Mathematics/dp/3110097451)

Answer by Buschi Sergio for Adjunction between Pro and Ind Completions

2012-07-24T13:50:41Z

No a answere but a specification and description of the issue (too long for a comment).

The free cocompletion property of $Ind(\mathcal{C})$ dont use the hypothesis "$\mathcal{C}$ has finite colimits". ANyway $Ind(\mathcal{C})$ is equivalent to the full subcategory $P_1(\mathcal{C})\subset \mathcal{C}^>$ of presheves that are coimits of a small, filtred diagram of representable. In this way the inclusion $i$ corresponds to the yoneda inclusion $h_-: \mathcal{C}\to P_1(\mathcal{C})$ and it preserves limit, and finite colimits. The inclusion $P_1(\mathcal{C})\subset \mathcal{C}^>$ create (small) filter colimits (if $P=\varinjlim_{i\in I}P_i$ is a filtrant colimit with $O_i\in P_1(P_1(\mathcal{C})$ then combining the comma categories $P_1(\mathcal{C})\downarrow P_i$ we make a filtrant (small) diagram of $P_1(\mathcal{C})\downarrow P_i$

Is $F: \mathcal{C}\to \mathcal{D}$ where $\mathcal{D}$ as filter colimits we have a (iso)unique extentions $F': P_1(\mathcal{C})\to \mathcal{D}$ with $F'(P):=\varinjlim_{\ (X, x)\in \mathcal{C}\downarrow P} F(X)$, if $P$ has the Ind-representation $(X_i)_{i\in I}$ i.e. $P=\varinjlim_i h_{X_i}$ then the diagram of the $h_{X_i}$ is a final diagram on the comma category $\mathcal{C}\downarrow P$ then $F'(P):=\varinjlim_i F(X_i)$

If $\mathcal{C}$ has finite colimits $P_1(\mathcal{C})\cong Cart(\mathcal{C}^{op}, Set)$ the latter is the the category cartesians presheaves i.e. that maps finite colimits of $\mathcal{C}$ to finite limits in $Set$, of course the embedding $Cart(\mathcal{C}^{op}, Set)\subset \mathcal{C}^>$ create limits .

All above as a dual version for $Proj(\mathcal{C}):=(Ind(\mathcal{C}^{op}))^{op}$, it is a free completion of $\mathcal{C}$, and if $\mathcal{C}$ has finite limits $Proj(\mathcal{C})$ is equivalent to $Cart(\mathcal{C}, Set)^{op}$ (dual to the category of copresheaves that preserving finite limits) it has (small) colimits and we have the embedding $\iota:=(h^-)^{op}: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op} $.

Now the inclusion $h_-: \mathcal{C}\to Cart(\mathcal{C}^{op}, Set)$ and $\iota: \mathcal{C}\to Cart(\mathcal{C}, Set)^{op}$ induce for the universal properties of these two completions the functors $U: Cart(\mathcal{C}, Set)^{op}\to Cart(\mathcal{C}^{op}, Set)$ with $U(Q)=\varprojlim_{\ h^Y\to Q} h_Y$, and $F: Cart(\mathcal{C}^{op}, Set)\to Cart(\mathcal{C}, Set)^{op}$ with

$F(P)=\varprojlim_{\ h_X\to P} h^X$ (the limit is in $\mathcal{C}^{<}$ the copresheaves category)

we have the natural isomorphisms:

$Cart(\mathcal{C}, Set)^{op}(F(P), Q)=Cart(\mathcal{C}, Set)(Q, F(P))=$

$\mathcal{C}^{<}(\varinjlim_{\ h^Y\to Q}h^Y, \varprojlim_{\ h_X\to P}h^X)=$

$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^<(h^Y, h^X)\cong$

$\varprojlim_{\ h_X\to P} \varprojlim_{h^Y\to Q}\mathcal{C}^>(h_X, h_Y)\cong$

$\mathcal{C}^>(\varinjlim_{\ h_X\to P} h_X, \varprojlim_{h^Y\to Q}h_Y) \cong$

$Cart(\mathcal{C}^{op}, Set)(P, U(Q))$

then $U$ is a adjoint to $F$.

About the notions of Grothendieck Universe and Tarski Universe

2012-07-04T22:06:36Z

I assume ZFC.

Let $U$ a set with the following (1), (2), (3):

1) $\omega\in U$

2) $x\in U\ \Rightarrow x\subset U$

3) $x\in U\ \Rightarrow \mathcal{P}(x)\in U$ (where $\mathcal{P}(x):=${$y| y\subset x$})

Then by these premises, I wish prove that $(4)\Leftrightarrow (4')$ where:

4) if $x\subset U$ and $|x|<|U|$ then $x\in U$ (where $|a|$ is the cardinality of the set $a$)

4') If $f: a\to U$ is function and $a\in U$ then $\bigcup_{s\in a}f(s)\in U$ .

The Book of Monk "Introduction to set theory' claim this equivalence as a exercise.

and the implication $(4)\Rightarrow (4')$ is immediate from the book above..

I tried to prove by induction that $|U|\subset U$, or tried to generalize the Mostowsky theorem for get a inijection $|U|\to U$ which preserves the relation '$\in $' (unsuccessfully). I have see also some posts here about this problem, but I hope exist a (relatively) simple answere..

THen I ask: How to prove $(4')\Rightarrow (4)$ ?

Answer by Buschi Sergio for Properties of functors and their adjoints

2012-07-02T11:02:21Z

THis is a resume from my old notes, the proofs aren't so difficult, but I include proof's if required....

PREMISES

Let $(F, G, \varepsilon , \eta): \mathscr{A} \to \mathscr{B}$ and adjunction.

Let $\Phi:{A, X}: (F(A), X)\cong (A, G(X)$ the natural bijection

give $f: F(A)\to X$ let $f^a:=G(f)\circ \eta_A$ its right adjoint give $g: A\to G(X)$ let ${}^ag:=\epsilon_X\circ F(f)$ its left adjoint

For $f: A\to A'$ da ${}^a(\eta_A'\circ f)=\epsilon_{ F(A')} \circ F(\eta_{ A'})\circ F(f)= F(f)$ follow that

$F_{ A, A'} = \Phi_{ A, FA'}^{-1} \circ \mathscr{A}(A, \eta_{ A'}): \mathscr{A} (A, A') \to \mathscr{A} (A, G(F(A'))) \cong \mathscr{B}(F(A), F(A'))$

THEN WE HAVE THE FOLLOWING PROPERTIES:

a) Give $G: \mathscr{C}\to \mathscr{A}$ let $\mathscr{A'} \subset\mathscr{A}$ the full subcategory with objects the $A\in \mathscr{A}$ such that $h^{A}_{G}: \mathscr{B}\to Set: B\mapsto (A, G(B))$ is representable

This is the maximum sub-category of which is defined a partial left adjoint $F$ of $G$, i.e. exist a bijection $\mathscr{C}(F(A), X)\cong \mathscr{A}(A, G(X))$ natural for $A\in \mathscr{A'}$ and $X\in \mathscr{B}$, then $F$ è unique but isomorphisms. Then $F$ preserves all colimits preserved by $\mathscr{A'} \subset_{fu}\mathscr{A}$ (also large or empty):

give a colimit cocone $(A_i\to A)_{i\in I} A_i$ in $\mathscr{A'}$ and a cocone $e_i: (F(A_i)\to X)_{i\in I}$ from the cocone $(e_i^a : A_i \to G(X))_{i\in I}$ follow unique $g: A\to G(X)$ with $g\circ \epsilon_i=e_i^a$ then ${}^ag: F(A)\to X$ is such that ${}^ag\circ F(\epsilon_i)=e_i$, if $g', g'' : F(A)\to X$ verify the last condition then $g'^a, g''^a : A\to G(X)$ are equal, then $g'={}^a(g'^a)= {}^a(g''^a)=g''$. Is easy proof that $F$ preserving epimorphisms, and dually $G$ preserving monomorphisms, and $F$ preserving strong.epimorphisms and dually $G$ preserving strong-monomorphisms.

b) The following properties are equivalent:

b.1) $F$ is faithful (full, full and faithful)

b.2) $\eta$ is a pointwise-monomorphism (pointwise-Retraction, a Isomorphism)

b.3) $F$ reflect monomorphism

b.4) $\Phi_{ A, B }$ preserving monomorphisms

b.5) For any $X\in\mathscr{C}$ the source $(a:X\to G(A))_{A\in \mathscr{A}, a\in (A, G(A))}$ is a mono-source (is enough considering $A$ belong to cogenerating class).

In Particular if $F$ is full from $1_G=G\varepsilon * \eta G$, $1_F= \varepsilon F*F\eta$ follow that $\eta G$, $G\varepsilon $, $F\eta$, $\varepsilon F$ are isomorphisms.

c) Here we call $F$ conservative is reflect isomorphisms, and call a morphisms $m: A\to B$ a co.cover if from $m=f\circ e$ with $e$ epimorphism follow that $e$ is a isomorphism, for straight generalization we have the definition of cocover source.

We have the implication: (1) $F$ is conservative $\Rightarrow $ (2) $F$ reflect co.Cover's $\Rightarrow $ (3) $\eta$ is pointwise-co.cover $\Leftrightarrow$ The source $(a:X\to G(A))_{A\in \mathscr{A}}$ is a co.cover source.

And $(3)\Rightarrow(1)$ if $F$ reflect isomorphisms on epimorphisms (I.e. if $F(e)$ is a isomorphism then $e$ is a epimorphism, in particular this happen if $F$ is faithful).

d) We call $F: \mathscr{B}\to \mathscr{A}$ co.fiathfull if for $H, K: \mathscr{A}\to \mathscr{C}$ and $\phi, \psi: H\to K$ and $\phi\circ F= \psi\circ F$ follow that $\phi=\psi$. ANd call $F$ co.conservative if (on the data above) from $\phi\circ F$ isomorphisms follow that $\phi$ is isomorphism.

We have the following equivalent properties:

d.1) $G$ if full and faithful

d.2) $\epsilon$ is isomorphism

d.3) $F$ is dense

d.4) $F\circ U$ is dense for some (any) $U: \mathcal{C}\to \mathscr{A}$ dense

d.5) the functor $F^*: \mathscr{B}[\Sigma]\to \mathscr{A}[\Sigma]$

where $\Sigma:=F^{-1}(Iso)$ , $F=F^*\circ P$, and $P: \mathscr{B}\to \mathscr{B}[\Sigma]$ canonic, is a equivalence

d.6) $F$ is co.fauthful $\Rightarrow$ $F$ is co.conservative.

e) G riflect strong.epimorphisms $\Leftrightarrow$ $\epsilon$ is pointwise-strong.epimorphisms

f) If $G$ is full and $\eta$ is pointwise-Section then $\eta$ is a Isomorphism.

g) Define a epimorphisms $e: X\to Y$ a (small)source-strong-epimorphism if give $f: X\to A$ and a (small) monosource $(m_i: A\to A_i)_{i\in I}$ and a (small) source $(g_i: Y\to A_i)_{i\in I}$ with $g_i\circ e=m_i\circ f\ i\in I$ exist unique a diagonal $d: Y\to A$ that keep the commutativity of the diagram.

We have te following property:

If for any $A\in \mathscr{A}$ the morphism $\epsilon_A : FG(A)\to A $ is (small)source-strong-epimorphism then $G$ reflect large (small) limits.\

h) Let $F$ such that for $X\in \mathscr{C}$ we have $1_X=s\circ r: X\to F(A)\to X$ for some $s,\ r$. From $\epsilon_X\circ FGF(r)=r\circ \varepsilon _{ FA }$ where $r$ and $\epsilon _{F(A)}$ retractions follow that $\epsilon_X$ is a retraction, then a epimorphisms and $G$ is faithful. If $G_{ A, A'}: \mathscr{B}(F(A), F(A'))\to \mathscr{A}(GF(A), GF(A'))$ is surjective then $G$ is full:

for $u: G(B_1)\to G(B_2)$ with $1=\rho_k\circ \sigma_k: A_k\to F(B_k)\to A_k$ follow $G(\sigma _2)\circ u\circ G(\rho_1): GF(B_1)\to GF(B_2)$ and this is $G(v)$ for some $v: F(B_1)\to F(B_2)$, then $u=G\sigma _2\circ v\circ Q\rho_1$.

Give the adjoint couples $(U_! , U^\ast)$ and $(U^\ast, U_\ast)$ where

$U^\ast: \mathscr{A}\to \mathscr{E}$.

For a category $\mathscr{C}$ let $\mathscr{C}^>:=Fun(\mathscr{C}^{op}, Set)$ the category of presheaves .We have the following equivalents properties:

i1) $U_!$ is faithfull and full (faithful).

i2) The unity $\eta_H: H\to U^\ast U_!(H)$, for $H\in \mathscr{A}^>$ is a isomorphisms (a monomorphism).

i3) $U_\ast$ is faithfull and full (faithful).

i4) The counity $\epsilon_H: U^\ast U_\ast (H)\to H$, for $H\in \mathscr{A}^>$ is a isomorphisms (a epimorphism).

Answer by Buschi Sergio for Defining ind-coherent sheaves and their singular support

2012-06-30T16:28:41Z

About the Ind completions of a category $\mathcal{C}$ :

Let $\mathcal{C}^>:=Fun(\mathcal{C}^{op}\to Set)$ the presheaves category.

we have the yoneda full embedding $Y: \mathcal{C}\to \mathcal{C}^>: X\mapsto h_X$.

For $P\in \mathcal{C}^>$ we indicate by $\int P$ the the comma category $\mathcal{C}\downarrow P$, with objects the couples $(X,x)$ with $X\in \mathcal{C},\ x\in P(X)$ and morphisms $f: (X, x)\to (Y, y)$ the $f\in \mathcal{C}(X, Y)$ such that $P(f)(y)=x$, we have the natural functor $\pi_P: \int P\to \mathcal{C}: (X, x)\mapsto X$ and is a foundamental theorem that $P\cong colim_{(X, x)\in \int P}\ h_X= colim\ Y\circ \pi $

we call $P$ flat if there is a final functor $\phi: I\to \int P$ (i.e. for any $\xi\in \int P$ the comma category $\xi\downarrow \phi$ is nonempty and connected) with $I$ a small filtered category (i.e. for any $\forall i, j\in I: \exists k\in I:\ I(i, k)\cup I(j, k)\neq\emptyset$ and for $\phi, \psi: I\to j$ exist $\theta: j\to k$ with $\theta\circ\phi=\theta\circ\psi$).

A flat functor $P$ preserve finite limit:

we can write $P\cong colim_{i\in I} h_{X_i}$ then

$P(Y)=colim_{i\in I}\mathcal{C}(Y, X_i)$ then if $Y= colim_{j\in J}$ then

$lim_{j\in J}P(Y_j)\cong lim_{j\in J}\ colim_{i\in I}\mathcal{C}(Y_j, X_i)\cong$ $\ colim_{i\in I}\ lim_{j\in J}\mathcal{C}(Y_j, X_i)\cong $ $colim_{i\in I}\mathcal{C}(colim_{j\in J} Y_j, X_i)\cong $ $colim_{i\in I}\mathcal{C}(Y, X_i)\cong P(Y)$.

Let $P_1(\mathcal{C})\subset\mathcal{C}^>$ the full subcategory of the flat presheaves. The category $P_1(\mathcal{C})$ has the small filtrant colimits (no too easy to prove).

Of course any representable is flat, then we have the Yoneda immersion $Y: \mathcal{C}\to P_1(\mathcal{C})$ A flat presheaf $P$ preserve finite colimits:

let $J$ a finite diagram, for any $P\in P_1(\mathcal{C})$ we have:

$(Y(colim_{j\in J}Y_j), P)\cong P(colim_{j\in J}Y_j)\cong$ $ lim_{j\in J}P(Y_j)\cong lim_{j\in J}(h_{Y_j}, P)\cong $ $(colim_{j\in J} h_{Y_j}, P)$

from Yoneda lemma follow that $colim_{j\in J}Y_j\cong colim_{j\in J} h_{Y_j}$ in $P_1(\mathcal{C})$.

We have the following universal property: let $F: \mathcal{C}\to \mathcal{A}$ where $\mathcal{A}$ has the filtered little colimits, then exist unique (to isomorphisms) the $F_1: P_1(\mathcal{C})\to \mathcal{A} $ preserving the small filtered colimits, with $F=F_1\circ Y$ (let $F_1:=Lan_{Y} F$ the left Kan extension of $F$ respect to $Y$).

Now, gived a $P\in P_1(\mathcal{C})$ we represent it as a filtered system $(X_i)_{i\in I}$

(with $P\cong colim_{i\in I}h_{X_i}$)

how we could represent a morfisms $\Phi: P\to Q$ (let fixed a representation $(Y_j)_{j\in J}$ of $Q$) ?

for $i\in I$ considering $h_{X_i}\to P\to Q\cong colim_{j\in J}h_{Y_j}$

by yoneda lemma, and the definition of colimit in $Set$, this has a factorization on some $f(i)\in J$ i.e. like some $\Phi_i: h_{X_i}\to h_{Y_{f(i)}}\to Q$

Then choosing for any $i\in I$ as above we have a map $f: I\to J$ and a famili o morphisms $(\Phi_i: X_i\to Y_{f(i)})_{i\in I}$, with the following property:

gived a $\phi: i\to i'$ we have a $k\in J$ and two morphisms $\psi: \Phi(i)\to k,\ \psi': \Phi(i')\to k$ such that $\psi'\circ\Phi_{i'}\circ X_\phi= \psi \circ \Phi_i$ (where $X_\phi: X_i\to X_{i'}$ is the canonical morphism).

Vice versa such data detecting a unique morphisms $\Phi: P\to Q$ that has such data as representation as above.

If $\Phi: P\to Q$ and $\Psi: Q\to R$ have the representation $(f, \phi)$ and $(g, \psi)$ (and $R\cong colim_{k\in K}h_{Z_k}$ then the composition $\Psi\circ \Phi$ is $(g\circ f, \psi\ast_{f, g} \phi)$ where $(\psi\ast_{f, g} \phi)_i: X_i\to Y_{f(i)}\to Z_{gf(i)}$.

The idetity $1_P: P\to P$ as the representation $(1_I, (1_: i\to i)_{i\in I}$.

Then we get a new category $Ind(\mathcal{C})$ equivalent to $P_1(\mathcal{C})$ with object the filtered diagrams in $\mathcal{C}$ like $(X_i)_{i\in I}$ and for morphism the representation $(f, \Phi)$ as above, with the above compositions and identities.

For the universal property above, follow a natural equivalence $P_1(P_1(\mathcal{C}))\cong P_1(\mathcal{C})$ then we have a natural equivalence $Ind(Ind(\mathcal{C}))\cong Ind(\mathcal{C})$.

Answer by Buschi Sergio for nontrivial isomorphisms of categories

2012-06-29T15:00:48Z

In my first step as student, I still do not know the concepts of category theory, but I thinked in "naive" way about this:

Let $Top$ the category of topological spaces and continuous maps (such that the invese images of opens are still opens), and let $Fil_T$ the category with objects like $(X, F)$ where $X$ is a set, $F=(F_x)_{x\in X}$ with $F_x$ a filter of subset with $x\in \bigcap F_x$ and such that $\forall U\in F_x\ \exists V\in F_x: \forall y\in V: U\in F_y$. ANd with morphisms $f: (X, F)\to (Y, G)$ , the maps $f_ X\to Y$ such that $\forall x\in X: \forall V\in G_{f(x)}: \exists U\in F_x: f(U)\subset V $.

The usual representation of topology as a family of opens or as families of neighbrhoods, and the essential identification between these, is just a isomorphisms between $Top$ and $Fil_T$ .

Of course this is a elementary example of isomorphism of concrete categories, in general a concrete isomorphism is just a different (but "equivalent") representation of the structure on a set, but these representation can have very different formalization (see all the literature on concrete categories, especially topological etc.).

Another nice example (I think) is the isomorphism (that fix the objects) between the Kleisli categori of a triple $T$ inducted from an adjuntion $(F, G): \mathcal{A}\to \mathcal{C}$ (right notation) and the clone category $\mathcal{C}_F$ of $F$ that has the some objects of $\mathcal{C}$ and with hom-sets defined as: $\mathcal{C}_F(X, Y)=\mathcal{A}(F(X), F(Y))$ (all these are considered disjoint) with the natural composition and identities of $\mathcal{A}$.

The following article is about a very nice isomorphisms (the shape category as the some objects and the shape functor fix these..):

http://www.jstor.org/discover/10.2307/1996811?uid=3738296&uid=2129&uid=2&uid=70&uid=4&sid=21100881872361

Answer by Buschi Sergio for Thompson's group F and monoidal categories

2012-06-27T21:23:46Z

Edit I noticed that in Fiore-Leinster preliminate the condition (free monoidal category of an isomorphism $ \alpha: A \otimes A \to A $) is different from what is written in the preliminary question, so I reworked my answer substantially.

In a Monoidal category $\mathcal{C}$ consider (a non empty) class of sections of the type $\beta: A\to A\otimes A$ and let $\Sigma$ its tensor product closure (finite tensor products of some morphisms of type $ \beta $ of the choose class and some identities).

From the article "Note on monoidal localizations " by B. Day (link text) the category of fraction $\mathcal{C}_\Sigma$ is (naturally) a monoidal category.

let $P: \mathcal{C}\to \mathcal{C}_\Sigma$ the natural functor.

The elements of $\Sigma$ are all monomorphisms (are sections) , and if $\Sigma$ admits a calculus of left fractions the canonical functors $P$ is faithful (see "Categories" H Shubert, 12.9.6(a), p.261). THen $\mathcal{C}$-$Aut(X)$ is a subgroup of $\mathcal{C}_\Sigma$-$Aut(X)$ (because $P$ is faithful).

Now consider the Monoidal category $[A, \alpha, \beta]$ free on (the condition):

"one object $A$ and on two morphisms $\alpha: A\otimes A\to A$, $\beta: A\to A\otimes A$, with $\alpha\circ \beta=1_A$".

This category has the following universal property: for any monoidal categories $\mathcal{C}$ with choose morphisms $a: X\otimes X\to X,\ b: X\to X\otimes X$ with $a\circ b=1_X$ there exists a unique strict monoidal functor $F_{a,b}: [A, \alpha, \beta]\to \mathcal{C}$ with $F(\alpha)=a,\ F(\beta)=b$.

Now in $[A, \alpha, \beta]$ consider the tensor closure $\Sigma$ of the section $\beta$,

and let $P:[A, \alpha, \beta]\to [A, \alpha, \beta]_\Sigma$ the category of fractions.

the category $[A, \alpha, \beta]_\Sigma $ has the universal property of the monoidal category on one isomorphisms

$\beta: A\to A\otimes A$ as in the FIore-Leinster article, then $F\cong [A, \alpha, \beta]_\Sigma$-$Aut(A)$.

Now, IF $\Sigma$ admit a calculus of left fraction then $P$ is faithful and $[A, \alpha, \beta]$-$Aut(A)$ is isomorphic to a subgroup of $F$.

P.S. I seems that $\Sigma$ admit a calculus of left fraction, but I have not checked it in detail

Answer by Buschi Sergio for Pseudofunctors out of the lax Gray tensor product

2012-05-17T18:51:50Z

This wants to be a track, I have not checked the details in their entirety

bibliography:

[G] J.W Gray Formal Category Theory: Adjointness for 2-Categories (Lnm 391).

Consider at first normal pseudo-funtors (on 2-cetegories) we call a pseudo.functor $F: \mathcal{A}\to\mathcal{B}$ normal if for any $A$: $F(1_A)=1_{FA}$ and the canonical isomorphism is the identity. Let $Fun_{np}(\mathcal{A}, \mathcal{B})$ the category of normal pseudofuntors and lax-transformations (with modifications too, is a 2-category). Now a normal pseudfunctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ give a family of normal pseudofunctors

$(F(A, -): \mathcal{B}\to \mathcal{C})_{A\in \mathcal{A}}$

$(F(?, B): \mathcal{A}\to \mathcal{C})_{B\in \mathcal{B}}$

such that $F(A, -)(B)= F(?, B)(A)$ and for $f: A\to A',\ g: B\to B'$ a 2-cell $\gamma_{f, g}$ (the $g$-component of the lax transformation $(F(f): F(A, -)\rightarrow F(A', -) $)

as in the diagram of [G] p. 57, which verify the properties $QF_21, QF_22,\ QF_23$ of of [G] p. 57. We call this "data" a normal quasi-pseudo-funtor. Similarly a 2-cell to induce what is called a lax transformation between normal quasi-pseudo-funtor.

mutually this data describe exactly a normal pseudofuctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ (this is sketched in [G] p. 60 for 2-functors) and this involve also lax-transformation, then we have a (isomorphism):

$Fun_{np}(\mathcal{A}, Fun_{np}(\mathcal{B}, \mathcal{C}))\cong n.q.p-Fun(\mathcal{A}\times \mathcal{B}, \mathcal{C})$

where the right member is the category (2-category) of normal quasi-pseudo-funtors and lax transformation (and modifications).

Now I think that exist a natural the isomorphism: $n.q.p-Fun(\mathcal{A}\times \mathcal{B}, \mathcal{C})\cong Fun_{np}(\mathcal{A}\otimes_w \mathcal{B}, \mathcal{C})$.

This follow as in [G] p.73,74,75, 76,77 (using also coherence criterion for pseudofunctor)

EDIT: the part about general pseudofunctors (no normal) , I'm working about...

Now consider general pseudo functors.

Let

[B] Introduction to Bicategories , J. Benabou.

Its enough show a natural equivalence $Fun_{pn}(\mathcal{A},\mathcal{B})\simeq Fun_{p}(\mathcal{A},\mathcal{B})$ where the latter member is the category of pseudo-functors and lax transformations.

These is a full inclusion $Fun_{pn}(\mathcal{A},\mathcal{B})\subset Fun_{p}(\mathcal{A},\mathcal{B})$ For $(F, \phi)\in Fun_{p}(\mathcal{A},\mathcal{B})$ give a natural construction of a 2-isomorphism (a lax transformation with isomorphisms components) $\eta_F: (F, \phi)\to N((F, \phi))$.

We have that $(F, \phi)$ consist of

a family of functors $F_{A,B}: \mathcal{A}(A, B)\to \mathcal{B}(FA, FB)$

a family of isomorphisms $\phi_A: I_{FA}\to F(I_A)$

a family of 2-isomorphisms $\phi_{f, g}: F(g)\circ F(f)\rightarrow F(g\circ f)$

with the usual coherence conditions M1, M2 p. 30 of [B].

Then define the normal $N((F, \phi))$ as $(F', \phi')$ where:

$F'(A):=F(A)$ for each object $A$ of $\mathcal{A}$ different form any $I_B$, and $F'(I_A):= I_{FA}$.

Let $F_{A, B}:= F_{A,B}$ for each couple of object $A,\ B$ of $\mathcal{A}$ both different form any $I_B$.

let $F'_{A, I_B}: \mathcal{A}(A, I_B)\xrightarrow{F_{A,B}}\mathcal{B}(F(A), F(I_B))\xrightarrow{(1, \phi_A^{-1})}\mathcal{B}(F(A), I_{F(B)}) $ (for $A$ different form any $I_B$)

Similarly we define $F'_{I_A, B}$ and $F'_{I_A, I_B}$.

Then let $\phi'_ A:=1: I_{F'A}\to F'(I_A)$

and define $\phi'_ {f, g}:= \phi_{f, g}$ if $g: B\to C$, $f: A\to B$ and each $A,\ B,\ C$ different form any $I_B$.

if for example only the codomain of $g$ is not of this type i.e. $g: B\to I_C$ then let

$\phi'_{f, g}: F'(g)\circ F(f)=\phi_C^{-1}\circ F(g)\circ F(f)\xrightarrow{\phi_C^{-1}\circ\phi_{f,g}} \phi_C^{-1}\circ F(g\circ f)=F'(g\circ f)$

Similarly we define $\phi'_{f, g}$ also if other of the objects $A,\ B, C$ are of type $I_D$ for some object $D$.

remains the verification of the conditions of consistency, but this follow from the general criterion of coherency for pseudo-functors (S. MacLane, R. Paré, "Coherence for bicategories and indexed categories") or for direct verification

Comment by Buschi Sergio

2013-04-29T11:12:57Z

Thank you. Searching about this question I just find this article, but I have to work more about it..

Comment by Buschi Sergio

2013-02-24T11:00:51Z

Dear Prof Holsztynski W, when I was student I studied your articles about shape theory (a categorical study) . By the way I remeber that for compact (T2 I seem) connected group the (weak) shape funcor is a equivalence. ams.org/journals/tran/1974-194-00/…

Comment by Buschi Sergio

2013-02-16T12:10:01Z

See B.Mitchell "Theory of Categories$ cpa. VII + Cor. 2.3 p.198.

Comment by Buschi Sergio

2013-02-10T14:36:05Z

If $F: \mathcal{A}\to \mathcal{B}$ is a functor then is naturally defined $Sk(F): Sk(\mathcal{A})\to Sk(\mathcal{B})$, then the inclusion $Cat_{Sk}\subset Cat$ induce $\iota: Sk(Cat_{Sk})\to Sk(Cat)$. We have a evident functor $j: Sk(Cat)\to Sk(Cat_{Sk})$ (from a small category take its skeleton, the its isomorphic element of $Sk(Cat_{Sk})$). We have $j\circ \iota=1$, $\iota\circ j\cong 1$. THen I find this construction, no very interesting.

Comment by Buschi Sergio

2013-01-23T18:53:17Z

your sequence isnt a Ker-Coker exat sequence like 0->(C, A)->(B, A)->(A, A)->0.

Comment by Buschi Sergio

2013-01-21T20:44:24Z

If you have $(f, 1): A\to (B, C)$ with $f: A\to B\otimes C$ and $(1, \sigma): (A, B)\cong (B, A)$ how do write the composition $(1, \sigma)\circ (f, 1)$ in the form $(\tau, g)$?

Comment by Buschi Sergio

2013-01-15T19:45:47Z

Please, can you explain how the first functor has a left adjoint which is a generalization of the Grothendieck construction? It is interesting.

Comment by Buschi Sergio

2013-01-06T13:04:01Z

Ok, I have the answere now.

Comment by Buschi Sergio

2013-01-06T12:59:50Z

I dont understand how a isomorphisms between identity endofunctors and other endofinctors are related to a existence of monoidal product and twist isomorphisms (partial defined symmetry).

Comment by Buschi Sergio

2013-01-05T19:49:43Z

Let $\mathcal{C}$ a category. Let $\mathcal{C}^>$ the category of presheaf on $\mathcal{C}$ and call $P\in \mathcal{C}^>$ a p-presheaf if it is isomorphic to a finite product of representable. Consider the full subcategory $\mathcal{A}\subset \mathcal{C}^>$ of the $P\in \mathcal{C}^>$ such that exist a epimorphism $e: \sum_{i\in I} P_i \to P$ from a (small) coproduct of p-presheaf to $P$. The inclusion $\mathcal{A}\subset \mathcal{C}^>$ create colimits and finite product, then $\mathcal{A}$ is cartesian monoidal and cocomplete (by product too). Is $\mathcal{A}$ closed?

Comment by Buschi Sergio

2012-12-29T10:45:23Z

E. Spanier "ALgebric TOpology", "Eilenberg Steenrod "ALgebric TOpology", GOdement "Topologie Algébrique et Théorie des Faisceaux ", COurant-Hilbert "Methods of Mathematical Physics"... "the problem of contemporary authors, is to being con-temporary" (Ennio Flaiano)

Comment by Buschi Sergio

2012-12-29T10:38:06Z

Engelking "General Topology"

Comment by Buschi Sergio

2012-12-17T08:38:49Z

Thank you, only I think all but this (simply) way . Grazie, ho pensato di tutto, tranne che questa via.

Comment by Buschi Sergio

2012-12-11T18:39:10Z

Isn't a well-placed question, is $v$ a funtor or a transformation between funtors $F$ and $G$?

Comment by Buschi Sergio

2012-12-08T18:03:59Z

Opps, I forget a comma "," between "symmetry" and "braids" in the title.