Definition df-termc 50939

Description: Definition of the proper class (termcnex 51045) of terminal categories, or final categories, i.e., categories with exactly one object and exactly one morphism, the latter of which is an identity morphism (termcid 50952). These are exactly the thin categories with a singleton base set. Example 3.3(4.c) of [Adamek] p. 24.

As the name indicates, TermCat is the class of all terminal objects in the category of small categories (termcterm3 50981). TermCat is also the class of categories to which all categories have exactly one functor (dftermc2 50986). See also dftermc3 50997 where TermCat is defined as categories with exactly one disjointified arrow.

Unlike https://ncatlab.org/nlab/show/terminal+category 50997, we reserve the term "trivial category" for (SetCat‘1_o), justified by setc1oterm 50957.

Followed directly from the definition, terminal categories are thin (termcthin 50943). The opposite category of a terminal category is "almost" itself (oppctermco 50971). Any category 𝐶 is isomorphic to the category of functors from a terminal category to the category 𝐶 (diagcic 51009).

Having defined the terminal category, we can then use it to define the universal property of initial (dfinito4 50967) and terminal objects (dftermo4 50968). The universal properties provide an alternate proof of initoeu1 18236, termoeu1 18243, initoeu2 18241, and termoeu2 50691. Since terminal categories are terminal objects, all terminal categories are mutually isomorphic (termcciso 50982).

The dual concept is the initial category, or the empty category (Example 7.2(3) of [Adamek] p. 101). See 0catg 17912, 0thincg 50924, func0g 50535, 0funcg 50531, and initc 50537.

(Contributed by Zhi Wang, 16-Oct-2025.)

Assertion

Ref	Expression
df-termc	⊢ TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

Distinct variable group:   𝑥,𝑐

Detailed syntax breakdown of Definition df-termc

Step	Hyp	Ref	Expression
1		ctermc 50938	. 2 class TermCat
2		vc	. . . . . . 7 setvar 𝑐
3	2	cv 1569	. . . . . 6 class 𝑐
4		cbs 17431	. . . . . 6 class Base
5	3, 4	cfv 6566	. . . . 5 class (Base‘𝑐)
6		vx	. . . . . . 7 setvar 𝑥
7	6	cv 1569	. . . . . 6 class 𝑥
8	7	csn 4588	. . . . 5 class {𝑥}
9	5, 8	wceq 1570	. . . 4 wff (Base‘𝑐) = {𝑥}
10	9, 6	wex 1812	. . 3 wff ∃𝑥(Base‘𝑐) = {𝑥}
11		cthinc 50883	. . 3 class ThinCat
12	10, 2, 11	crab 3416	. 2 class {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}
13	1, 12	wceq 1570	1 wff TermCat = {𝑐 ∈ ThinCat ∣ ∃𝑥(Base‘𝑐) = {𝑥}}

This definition is used by:  istermc  50940