Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A theory of 2-pro-objects, a theory of 2-model 2-categories and the
  2-model structure for 2-Pro(C)   In the sixties, Grothendieck developed the theory of pro-objects over a
category. The fundamental property of the category $Pro(C)$ is that there is an
embedding $C \stackrel{c}{\rightarrow} Pro(C)$, $Pro(C)$ is closed under small
cofiltered limits, and these are free in the sense that for any category $E$
closed under small cofiltered limits, pre-composition with $c$ determines an
equivalence of categories $Cat(Pro(C),E)_+ \simeq Cat(C,E)$, (the $+$ indicates
the full subcategory of the functors that preserve cofiltered limits).
  In this work we develop a 2-dimensional pro-object theory. Given a 2-category
$\mathcal{C}$, we define the 2-category $2$-$\mathcal{P}ro(\mathcal{C})$ whose
objects we call 2-pro-objects. We prove that
$2\hbox{-}\mathcal{P}ro(\mathcal{C})$ has all the expected basic properties
adequately relativized to the 2-categorical setting, including the
corresponding universal property. We give an adecuate definition of closed
2-model 2-category and demonstrations of its basic properties. We leave for a
future work the construction of its homotopy 2-category. Finally, we prove that
our 2-category $2\hbox{-}\mathcal{P}ro(\mathcal{C})$ has a closed 2-model
2-category structure provided that $\mathcal{C}$ has one.
  Part of the motivation of this work was to develop a conceptual framework to
handle the $\check{C}$ech nerve in homotopy theory, in particular in strong
shape theory. The $\check{C}$ech nerve is indexed by the categories of covers
and of hypercovers, with cover refinements as morphisms, which are not filtered
categories, but determine 2-filtered 2-categories on which the $\check{C}$ech
nerve is also defined, sends 2-cells into homotopies, and determines a
2-pro-object of simplicial sets. Usually, the $\check{C}$ech nerve has to be
considered as a pro-object in the homotopy category, loosing the information
encoded in the explicit homotopies.

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The categories of this paper are: funct-an
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
7
The categories of this paper are: hep-th
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The categories of this paper are: acc-phys
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The categories of this paper are: q-alg
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The categories of this paper are: math-ph
6
The categories of this paper are: astro-ph
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The categories of this paper are: eess
11
The categories of this paper are: hep-lat
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The categories of this paper are: stat
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The categories of this paper are: cs