Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Higher Equipments, Double Colimits and Homotopy Colimits   This document is centered around a main idea: simplicial categories, by which
we mean simplicial objects in the category of categories, can be treated as a
two-fold categorical structure and their double category theory is
homotopically meaningful. The most well-known two-fold structures are double
categories, typically used to organize bimodules in various contexts. However
there is no double category of spaces even though notions of bimodule are
conceivable. We first remedy this defect of double category theory by
constructing a meaningful simplicial category of spaces. Then we develop the
analogy with double categories by defining double colimits and by postulating
an equipment property, which is promptly satisfied in the examples. As an
application we prove that certain double colimits are naturally interpreted as
homotopy colimits. Quite surprisingly this analogy unveils a principle:
simplicial categories are to simplicially enriched categories what double
categories are to 2-categories!

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The categories of this paper are: supr-con
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The categories of this paper are: alg-geom
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The categories of this paper are: q-fin
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: quant-ph
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The categories of this paper are: adap-org
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The categories of this paper are: stat
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: solv-int
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The categories of this paper are: q-alg
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The categories of this paper are: eess