Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Eilenberg-Watts theorem in homotopical algebra   The object of this paper is to prove that the standard categories in which
homotopy theory is done, such as topological spaces, simplicial sets, chain
complexes of abelian groups, and any of the various good models for spectra,
are all homotopically self-contained. The left half of this statement
essentially means that any functor that looks like it could be a tensor product
(or product, or smash product) with a fixed object is in fact such a tensor
product, up to homotopy. The right half says any functor that looks like it
could be Hom into a fixed object is so, up to homotopy. More precisely, suppose
we have a closed symmetric monoidal category (resp. Quillen model category) M.
Then the functor T_{B} that takes A to A tensor B is an M-functor and a left
adjoint. The same is true if B is an E-E'-bimodule, where E and E' are monoids
in M, and T_{B} takes an E-module A to A tensored over E with B. Define a
closed symmetric monoidal category (resp. model category) to be left
self-contained (resp. homotopically left self-contained) if every functor F
from E-modules to E'-modules that is an M-functor and a left adjoint (resp. and
a left Quillen functor) is naturally isomorphic (resp. naturally weakly
equivalent) to T_{B} for some B. The classical Eilenberg-Watts theorem in
algebra then just says that the category of abelian groups is left
self-contained, so we are generalizing that theorem.

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The categories of this paper are: nlin
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The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: econ
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The categories of this paper are: hep-ex
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
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The categories of this paper are: eess
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int